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Muonium chemistry in condensed media Ng, Chi Biu William 1983

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C . I MUONIUM CHEMISTRY IN CONDENSED MEDIA B.Sc, The University of British Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES (Department of Chemistry, U.B.C.) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA OCTOBER, 1983 @ CHI BIU WILLIAM NG, 1983 by CHI BIU WILLIAM DOCTOR OF PHILOSOPHY i n In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of CtfBWS 7& X  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Oc% , A / . / » £ 3  DE-6 (3/81) ABSTRACT Muonium (y +e , chemical symbol Mu) consists of an orbital electron associated with a positive muon as nucleus. It can be regarded as a very light 'isotope' of the hydrogen atom because i t has essentially the same Bohr radius and ionization energy. Thus i t can be used as a sensitive probe of isotope effects and of H-atom reactions which cannot be studied by conventional techniques. Due to the unique nuclear spin properties of the muon, there are several techniques available for investigation. These include muon spin rotation (ySR), muonium spin rotation (MSR) and muonium radical spin rotation (MRSR) in transverse magnetic fi e l d s , as used in this study. Various fundamental aspects of muonium formation and of chemical reaction kinetics have been explored by the experiments presented in this thesis. These are summarized below. (i) From the magnetic f i e l d dependence, i t was verified that Mu does not react chemically with water to any significant extent. Its observed spontaneous slow spin relaxation arises from experimental artifacts such as magnetic f i e l d inhomogeneities and/or Mu-frequency beating. ( i i ) The MRSR technique was used to observe and identify muonium-substituted free radicals via their pair of precession frequencies in high transverse magnetic fields in pure benzene, pure styrene, and their mixtures. The results have implica-tions regarding the mechanism of radical formation and selectivity, ( i i i ) Both ySR and MSR experiments were performed on neopentane (liquid § solid) and concentrated KOH solutions. The y+ and Mu yields in these systems indicated that a spur model of Mu formation is neither appropriate nor adequate to explain the results, (iv) In muonium solution kinetic studies, the reaction Mu + OH was found to be relatively slow, with a substantial a c t i -aq ' vation energy (E ) and no kinetic isotope effect compared to H at room temperature. The reaction shows Mu behaving as a "muonic" acid, (v) Kinetic studies of the abstraction of D by Mu from D C O 2 as a solute in water gave a large E^. Upon comparison with HC^", the isotope effects (k^/k^ and k^/k^) imply that quantum mechanical tunnelling does not dominate the abstraction of H and D atoms in H C 0 2 ~ and D C 0 2 ~ by either H or Mu at room temperature, (vi) The MSR technique was used to i n i t i a t e a study of model biological systems (various solutes incorporated in cyclodextrins and micelles). The results demonstrated the sensitive and non-destructive nature of the MSR technique, (vii) Hydrocarbons were also investigated: including measuring their muon yields, their temperature dependence, the effect of an external electric f i e l d , and yields in solvent mixtures. Almost a l l the data obtained seem to be at variance with the expectations of significant intra-spur processes in Mu formation, but are consistent with that of a 'hot atom' mechanism. -iv-TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS iv LIST OF TABLES v i i i LIST OF FIGURES x ACKNOWLEDGEMENT x i v CHAPTER 1. INTRODUCTION 1 l.A. Muonium, Past and Present 2 l.B. Models of Muonium Formation 4 I.C. Reaction Dynamics in the Liquid Phase 6 l . C . l . Types of Reactions in Solutions 8 I.C.2. Transition State Theory Applied to Liquid Phase Reactions 12 1. C.3. Isotope Effects and Quantum Mechanical Tunnelling 15 1. D. Description of Thesis Content 19 CHAPTER 2. THEORY AND EXPERIMENTAL METHOD 21 2. A. Theory of the Experimental Method 22 2. A.I. Muon Production 22 2.A.2. Muon Polarization 23 2.B. Experimental Techniques 24 2.B.I. The ySR Technique 27 2.B.2. The MSR Technique 28 -V-TABLE OF CONTENTS (cont'd) Page 2.B.3. The MRSR Technique 33 2.C. Electronic Logic and Experimental Set-up . . . . 35 2.C.I. Surface Muon Set-up 38 2.C.2. Backward Muon Set-up 40 2.D. Sample Preparation and Target-Holders 42 2.D.I. Surface Muon Sample Holders 42 2.D.2. Backward Muon Sample Holders 47 2.E. Data Analysis 47 2.E.I. Analysis of pSR and MSR Spectra . . . . 49 2. E.2. Analysis of MRSR Spectra 54 CHAPTER 3. ORIGIN OF A IN WATER: A MAGNETIC FIELD DEPENDENCE STUDY . . °. 57 3. A.I. Results 62 3. A.2. Discussion 66 CHAPTER 4. MUONIUM RADICALS IN BENZENE, STYRENE, AND ITS MIXTURES 71 4. A.I. Results 74 4.A.2. Discussion 80 4. A.3. Conclusion 87 CHAPTER 5. MUON AND MUONIUM YIELDS 89 5.A. Muonium Atoms in Liquid and Solid Neopentane . . 90 5. A.I. Results 91 5.A.2. Discussion 91 5.B. Muon Yields in Concentrated OH" Solutions . . . 99 - v i -TABLE OF CONTENTS (cont'd) Page 5.B.I. Results 99 5. B.2. Discussion 100 5. C. Conclusion 107 CHAPTER 6. MUONIUM KINETICS IN AQUEOUS SOLUTIONS 109 6. A. Muonium Reaction with OH 110 aq 6. A.I. Results I l l 6.A.2. Calculations and Discussion 116 6. B. Muonium Abstraction with D C O 2 120 6.B.I. Results 120 6. B.2. Discussion 121 CHAPTER 7. COLLABORATIVE WORKS 129 7. A. MSR Applications to Model Biological Systems . . 130 7. A.I. Muonium Reactivity in Cyclodextrins . . 131 7.A.2. Muonium Reactivity in Micelles 134 7.B. Studies with Hydrocarbons 138 7.B.I. Yields in Liquid Hydrocarbons 138 7.B.2. Temperature Dependence of Muonium in Hydrocarbons 142 7.B.3. Effect of External Electric Fields on the ySR of Liquid Hydrocarbons and Fused Quartz 146 7.B.4. ySR Studies with Solvent Mixtures . . . 150 7.C. Muonium Solution Kinetics 153 7.C.I. Muonium Addition to Vinyl Monomers . . . 154 - v i i -TABLE OF CONTENTS (cont'd) Page 7.C.2. Spin Conversion Reaction of Muonium with Nickel (II) Cyclam 158 7.C.3. Muonium Addition to Cyanides 161 CHAPTER 8. SUMMARY AND CONCLUSION 165 REFERENCES 172 APPENDIX I. CORRECTIONS FOR MU-RADICAL AMPLITUDES "... 180 I.a. Effect of Timing Resolution of the Detector System on the Amplitude of Sinusoidal Oscillations 181 I.b. Effect of Packing Factor on Amplitudes 187 I. e. Other Effects 190 APPENDIX II. RESIDUAL POLARIZATION OF A SINGLE-STEP MU REACTION MECHANISM 191 II. a. Residual Polarization i n Liquids 192 II.b. Implication of Mu Reaction Kinetics on P r e s • • • 193 II.c. The Ensemble of Muon Polarization in a Single-Step Mu Reaction Mechanism 194 Il.d. The General Expression of P r e s for A l l Fields . . 195 II.e. Different Field Limits of P 196 II. e . l . B < 10G 196 II.e.2. 10G < B < 150G 197 II.f. Inclusion of Hot-Atom Reactions 198 II.g. Application of P to Muon Yields in Concentrated OH" Solutions . T.es. 199 - v i i i -LIST OF TABLES Table £ M £ 1.1 Physical Parameters of the Muon and the Muonium Atom . . 2 2.1 Typical Asymmetry Values for Backward and Surface Muons 49 3.1 X Values Obtained by Fitting with and without Consideration of Mu-degeneracy at Low Magnetic Fields . 64 4.1 Hyperfine Coupling Constants of Styrene and Benzene . . 76 4.2 Radical Frequencies at 3.4 kG in Styrene (S^ and S„) and Benzene (B^ and B^) and Correction Factors Needed to Calculate the Amplitudes 78 4.3 Various k.. and k„ Values for Benzene and Styrene . . . . 86 M H 5.1 Values of X , PM, P n and P. in Neopentane in Liquid and Solid Pnases . 7 . . . 93 5.2 Physical Properties Affecting Electron Escape from Spurs, and P^ , P Q and P L Values Obtained for Four Pure Liquids at 295 K 94 5.3 The Effect of Phase (and Temperature) on the Various Muon Yields in Neopentane, Water and Argon 96 5.4 Variation of k M or h Required for Equation (5.9) to Fit the Experimental P n Data 108 6.1 Second Order Rate Constants (k M) Obtained from X M Using Equation (2.16) at Various Temperatures and OH" Concentrations 113 6.2 Arrhenius Parameters Obtained for k^ at ~ 295 K, and Comparison with: (i) k H, and ( i i ) Reaction of Mu with HC02" 115 6.3 Viscosity Parameters as a Function of Temperatures for the Reaction of Mu with OH " 117 aq 6.4 Muonium Decay Rate Constants ( X^ ) as a Function of DC02~ Concentration at 80°C 122 - i x -LIST OF TABLES (cont'd) Table Page 6.5 Second Order Rate Constants (kw) Obtained from Using Equation (2.16) at Four Temperatures f o r the Mu + DC0 2" Reaction 124 6.6 Comparative K i n e t i c Parameters f o r Various Abstraction Reactions 126 7.1 Muonium Reaction Rate Constants (k^) with and i n Various Solutions at 295 K 133 7.2 Reaction Rate Constants k^/lO"^ M~*s~* of Muonium in Water, an Organic Solvent, and i n NaSOA M i c e l l a r Aqueous Solutions, at 295 K 136 7.3 Results Obtained f o r P Q , Vy[, P L a n d i n n-Hexane, c-Hexane, Tetramethylsilane (TMS), Methanol and Water 139 7.4 Results Obtained f o r X i n Solutions 140 7.5 Comparison of P^ and (1-PQ ) with Properties o f the Liquid: (i) the Fraction of Electrons which Escape Intraspur N e u t r a l i z a t i o n i n Low LET Radiolysis ( G ^ ^ / G t ) ; ( i i ) the Electron M o b i l i t y ( y ); and ( i i i ) the S t a t i c D i e l e c t r i c Constant (e) 143 7.6 E f f e c t of EEF on A D and A M i n Fused Quartz 148 7.7 E f f e c t of EEF on A D i n Various Liquids 149 7.8 Hyperfine Coupling Constants 155 7.9 Reaction Rate Data of Monomers at 295 K 157 7.10 K i n e t i c Data of Cyanide System with Some Associated Mu Rate Constants 163 -X-Figure 1.1 1.2 1.3 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 LIST OF FIGURES Page Diagrams Showing the Contrasting HOT and SPUR Models for Muonium Formation in Liquids 7 (a) Diagram of the Reaction Path Through the Energy Barrier 14 (b) Curve of the Potential Energy Plotted Against the Reaction Coordinate (after Entelis and Tiger [66]) 14 The Dependence of a Wave Function (ty) on the Potential 17 Schematic Diagram Showing the TRIUMF Site at U.B.C. . . 25 Schematic Diagram Showing the Set-up of Quadruple and Bending Magnets for the M20-Muon Channel at TRIUMF 26 Time Histograms Showing the Field Dependence of the Diamagnetic Precession Signal at (a) 100G and (b) 1500G for Target Solution of CC£ 4 29 "Raw" and "Asymmetry" MSR Histograms of Water 31 Breit-Rabi Diagram for a Two-spin-% System 32 Muonium Radical in Styrene Showing (a) the Raw Histogram at 1500G, and FFT Spectra at (b) 1500G and (c) 2500G 34 FFT Spectra Showing Precession Frequencies of the Three Techniques (ySR, MSR, MRSR) 36 Simplified Schematic Diagram Showing Data Acquisition System at TRIUMF 37 Diagram of Surface Muon Apparatus Geometry 39 Experimental Set-up for Backward Muon Experiments . . . 41 Side View of (a) Regular Teflon Cell, and (b) Temperature Teflon Cell for Surface Muon Experiments . 43 - x i -LIST OF FIGURES (cont'd) Figure Page 2.12 Schematic Diagram of the Surface Muon Temperature-Dependence Experimental Set-up 45 2.13 Ty p i c a l MSR Asymmetry Histograms Showing the Decay of Muonium Signal at 9 gauss f o r 5 x 10" 5 M KMnO, at (a) 3°C, (b) 58°C, and f o r (c) 3.5 x 10" 4 M NaN03, (d) 1.4 x 10" 3 M NaN03, both at 1°C 52 2.14 Plot of Observed Mu-decay Constants ( X M ) Against Solute Concentrations (Hemin and Protoporphyrin) at Room Temperature to Obtain 53 2.15 FFT Diagrams Showing Mu-radical Formation i n Neat Styrene and Neat Benzene at 3400G 55 3.1 MSR Measurement of Mu in Quartz at 17G 61 3.2 MSR Histograms of Mu i n Water F i t t e d with Equation (3.7) at (a) 3.85G, (b) 6.44G, and (c) 9.38G 63 3.3 Magnetic F i e l d Dependence of AQ f o r Water 65 3.4 Diagram of Expected X Dependence vs. [S] F i t t e d with (a) Equation (2.6) and (b) Equation (3.7) 69 4.1 FFT Spectrum of 20% Styrene by Volume i n Benzene Obtained from the 3.4 kG MRSR Histogram 77 4.2 Plot of Normalized, Corrected Radical Y i e l d s Against the Volume-Percent of the Benzene/ Styrene Mixture 79 4.3 Plot of Total Yields (PQ^PR § p j J Against Volume Fracti o n of the Benzene-Styrene Mixtures 81 4.4 K i n e t i c Competition Plots 83 5.1 Raw Histograms (dots) and Computer F i t s ( l i n e s ) f o r Liquid Neopentane at 295 K; (a) Showing the Muonium Precession with 8G Transverse Magnetic F i e l d ; and (b) Showing the Diamagnetic Muon Precession at 80G 92 - x i i -LIST OF FIGURES (cont'd) Figure Page 5.2 Variation of P D with Composition of KOH/r^O Solutions . 101 5.3 Plot of P n Against log [KOH] 106 6.1 Typical MSR Histograms Showing the Muonium Signal at 8 gauss for (a) Pure Water at Room Temperature and (b) Its Decay at 0.01 M KOH at 59°C 112 6.2 Arrhenius Plot Showing ln(k M) versus T"1 for the Mu+0H~ Reaction 114 6.3 Plot of k M/T versus the Inverse of the Viscosity for the Mu+OH- Reaction 118 6.4 Plot of X M versus DC02" Concentration at 80°C 123 6.5 Arrhenius Plot of the Mu+DC02" Reaction 125 7.1 Muonium and Diamagnetic Muon Parameters for Cyclo-hexane from Room Temperature Down to -150°C 144 7.2 Muonium and Diamagnetic Muon Parameters for n-Hexane from Room Temperature Down to -150°C 145 7.3 Plots of P n against Volume Fraction of CC£^ or Cyclohexane in Benzene (1) for CC^-Benzene Mixtures and (2) for Cyclohexane-Benzene Mixtures 151 7.4 Plot of the Observed Rate Constant, given as (*A-A0)/[Ni], as a Function of the Fraction of Nickel-Complex Species in the Paramagnetic State (Fp) . . . . 160 1-1 Timing Resolution Curve of Positronium System by Measuring the Two y' s Emitted from a 60co Source . . . 182 1-2 The Effect of Timing Resolution (2x or At) on a Sinusoidal Function 185 1-3 T 0-Calibration or Timing Resolution Curve of the Forward Histogram for Backward Muon Set-up 186 - X l l l -LIST OF FIGURES (cont'd) Figure 1-4 Timing Resolution Curve of Backward Muon Set-up Using Quartz as the Emperical Measurement . . . 1-5 E f f e c t of B i n s / O s c i l l a t i o n Period on Amplitude -xiv-ACKNOWLEDGEMENT It is with the greatest pleasure to thank my research supervisor, Dr. David C. Walker, who has been both a very good friend and a tremendously patient advisor, for his helpful guidance and enlightening discussion. I must also give sincere thanks to Drs. Y.C. Jean (Univ. of Missouri-Kansas City) and T. Suzuki (Hachinole Institute of Technology, Japan) for being my experimental teachers during my early days at TRIUMF. My whole-hearted gratitude must go to the delightful collaborations with Drs. Y. Ito (Univ. of Tokyo) and J.M. Stadlbauer (U.B.C. and Hood College, Maryland) during my graduate years. Furthermore, to Y. Miyake (Univ. of Tokyo), Dr. T. Nguyen (Univ. of Zurich), and R. Ganti (UMKC), their help during TRIUMF experiments are deeply appreciated. Of course, this thesis could not have been completed without the many TRIUMF colleagues (J.H. Brewer, D.G. Fleming, D.M. Garner, R. Kief1, G. Marshall, R.J. Mikula, and D.P Spencer). I am indebted to their many useful suggestions. Finally, I must give a l l my thanks to my parents, whose support and patience provided me with energetic enthusiasm. It is with great respect that I dedicate to them this Ph.D. thesis. -1-CHAPTER 1 INTRODUCTION -2-l.A. Muonium, past and present. Muonium (chemical symbol, Mu) is the atom consisting of a positive muon and an electron. The electron orbits the muon as i t s nucleus, therefore forming an analogous hydrogen atom. The muonium atom is viewed as the simp-lest system since both the muon and the electron are leptons [ l ] , whereas the hydrogen atom (H) can now be considered as a four-particle system due to the advent of the theory of quarks [2], In any event, chemically, the Mu atom can be regarded as an ultralight, radioactive isotope of the H atom. This is i n t r i n s i c a l l y obvious as the following table outlining the physical properties of the muon and Mu atom indicates [3,4,5]. Table 1.1 Physical parameters of the muon and the muonium atom. Muon: Spin Mass Mean lifetime Magnetic moment Muonium: Bohr radius Ionization potential Hyperfine oscillation period 1/2 M = 206.77M = 0.11261M y e p x = 2.1971xl0"6 sec y p = 3.1833y = 4.835x10"3y V P c = 0.5317 A = 1.0043a (H) I = 13.539eV = 0.9957(H) P 2.24xl0" 1 0 sec -3-The Bohr radii and the ionization potentials of both H and Mu are almost the same. In this sense, Mu is a true isotope of H. Its collisions with other atoms and molecules, especially chemical reactions, can be studied to examine -with improved sensivity (relative to deuterium and tritium) such aspects as kinetic isotope effects and quantum mechanical tunnelling by comparing reac-tions of Mu with those of H. In 1957, Garwin et al. [6] and Friedman et al. [7] made the f i r s t experi-mental observations of positive muons after the verification of the break-down of the principle of parity invariance [8]. Although these two groups employed slightly different apparatus, their methods were forerunners of the o present day ySR (muon spin rotation) technique. In the next few years, Hughes et al. [9] discovered free Mu atoms in argon gas and subsequently acclaimed i t as an ultralight isotope of H atom [lO]. Over the next decade, important experimental foundations of muon chemistry were paved by scientists studying the physical aspects of the muon and the muonium atom [11,12,13,14, 15]. In addition, a series of theoretical papers by Soviet groups led the way to fundamental understanding of the reaction and formation of Mu atoms [16,17,18]. During the 1970 s, continuing experiments including measure-ments of the muon magnetic moment [19], the muon anomalous magnetic moment [20], and the hyperfine s p l i t t i n g of the Mu atom [1,21], helped to stablize and expand the f i e l d of muon and muonium chemistry immensely. Muonium chemistry in condensed media was f i r s t studied by a group at Dubna [11,12,13] and then more extensively by a group at the Lawrence Berkeley Laboratory (LBL) [4,22,23]. It was not however unt i l 1976 that -4-the f i r s t direct observation of Mu in water was achieved by Percival et al. at the Swiss Institute for Nuclear Research (SIN) near Zurich [24], Since then, this group has continued to produce impressive data [25], including -results in muonium-substituted transient radicals [26]. Here at TRIUMF, before 1978, the major investigation of Mu chemistry was centered around gas-phase studies [27,28]. However, since then, Jean et al. [29,30,31,32] have started and maintained a very competitive program of Mu chemistry in the condensed phase. In addition, recent direct observation of Mu in various hydrocarbons was successfully carried out at TRIUMF [33,34,35]. The applications of Mu as a nuclear probe in large macromolecular and biological systems [36,37,38,39,40] have also been a major concern here at U.B.C. Lastly, the recent use of Mu and Mu-substituted free radicals to study vinyl monomers [41] could give new insights about polymer i n i t i a t i o n kinetics. As one can see, the f i e l d of Mu chemistry in the condensed phase is fast becoming of broad prospective and reaching into several conventional areas of chemical study. l.B. Models of muonium formation. At present, there are two models of muonium formation. One is referred I I n I I n to as the HOT model [4], while the other i s called the SPUR model [42]. In the hot model, which was conceptualized earlier by physicists, essen-t i a l l y views an epithermal muon - with an energy up to several keV - abstrac-ting an electron from the medium so that Mu is formed beyond the t r a i l of ionizations (spurs) of the muon track. During i t s thermalization a certain -5-fraction of Mu reacts with the medium, some of which give diamagnetic muonic species or stay as free muons (P ), some reach thermal energy as free Mu atoms (P^) or as muonic free radicals (P^), and some simply remain unobser-vable or lost (P^) [43,44]. These fractions represent the relative proba-b i l i t i e s of these reactions involved in the relevant energy range extending from several MeV, at which the muon enters the medium, down to thermal energies. However, in condensed media, these reaction probabilities are extremely d i f f i c u l t either to calculate or estimate; therefore making the hot model a very general and yet vague theory of muonium formation. In radiation chemistry, a t r a i l of spurs is a result of energy deposi-tion to the medium from the passage and thermalization of a high energy charged particle, such as a positive muon. Typically, these spurs are regarded to result from the deposition, on average, of about lOOeV of energy in a sphere of about 2 nm diameter in water. Therefore, these spurs are minute regions in the particle s track containing high concentrations of very reactive free radicals, electrons, positive ions, and excited mole-cules [45]. In positronium studies, Mogensen has postulated a spur model in which he argues that the positron Ce+) reaches thermal energy while s t i l l in the neighbourhood of reactive free radicals, electrons and ions generated in spurs in i t s thermalization track [46], Recently, this has prompted an analogous SPUR model of Mu formation by Percival et al. L42J. The model pictures a muon being thermalized within the last spur of i t s own track in which i t can either encounter a radiation-produced electron to become Mu or i t can escape into the medium; at the same time, Mu can either perform intra--6-spur reactions (including fast depolarization by a secondary solvated electron [47]) or escape from the spur. Therefore, this model attempts to explain the i n i t i a l distribution of muons between diamagnetic and paramagnetic species -simply based on the relative probabilities of reaction or escape of therma-lized muons from the reactive species of the terminal spur. Although the spur model is more specific than the hot model, i t is by no means a better theory of muonium formation. So far, there have certainly been more argu-ments against this spur model [28,33,34,35,48,49,50,51] than for i t [43,44 52,53]. The two contrasting models, HOT and SPUR, have been in direct conflict for the proper mechanism of muonium formation in liquids. At this moment, both models are s t i l l under constant criticisms and investigations from the muon community and i t is not yet so clear how Mu is formed, although the general consensus is that i t s formation could be some fractional contribu-tions from both the HOT and SPUR theories. The schematics of the two models are illustrated in Figure 1.1. In the drawing, the time scales are mere differentiating guidelines between the two mechanisms and that the times of formation are only estimated. I.C. Reaction dynamics in the liquid phase. Contrary to gas phase studies, our knowledge of molecular motion and of molecular energy levels in liquids is not well developed. Therefore our approach toward encounters of molecules in solutions is quite different to that in gases. For example, gas molecules at STP occupy about 0.2% of Figure 1.1: Diagrams showing the contrasting HOT and SPUR models for muonium formation in liquids. -8-the total volume while in the liquid this same figure rises to about 70% or more. This difference means that, while motion in gases is largely unhin- '. dered, molecules in solutions have to jostle their way past one another in .order to make any substantial displacement. Clearly, the encounter frequency between two specified molecules i n i t i a l l y separated is much smaller in solu-tions than that in gases. However, there is an important factor in this difference. In solutions, since molecules diffuse slowly into the region of a possible reaction partner, they also migrate slowly away from i t . There-t fore, once partners of a reaction move into each other s v i c i n i t y - into the M i t same solvent cage - they collide with one another many times before escap-ing this solvent cage. In other words, once an encounter has occurred, the chance of undergoing a reaction is greatly enhanced compared to the gas phase. As a matter of fact, the time spent by a molecule in one of these solvent cages can be as long as 100 psec; and during this period, the molecule can make between 100 to 1000 collisions with i t s neighbouring molecules [54]. l . C . l . Types of reactions in solutions. In order for molecule A to react with molecule B they must f i r s t come into contact with each other, and this requires the molecules to be in the M I I same solvent cage where they are then referred to as the encounter pair [55]. The activation energy of a reaction (E ) is now a complicated quantity (relative to gases) because the encounter pair is surrounded by solvent molecules and i t s reaction path is determined by a l l the neighbouring inter-actions. However, this situation can be depicted by the following reaction scheme: -9-A + B ±{A...B} P (1.1) where {A...B} is the encounter pair, k^ and k^ are the diffusive-approach and separation rate constants, respectively, and k^ is the reaction rate constant giving product P. The steady state concentration of {A...B} can be found from the following rate equation: d [ ^ B } ] = k d[A][B] - kb[{AB}] - kp[{AB}] = 0 (1.2) which gives: [{AB}] = [A][B] (1.3) Substituting equation (1.3) into the rate expression for the formation of P gives the overall equation as: 4fi. y U B , ] . k k. p d k,+k b p [A][B] (1.4) In equation (1.4) the overall rate law is second-order, and the bimolecular -10-rate constant, (experimentally observable, k Q ^ s ) , is given as: k k, p d k. = . (1.5) b p From equation (1.5), two limits can be recognized. If the rate of breakup of the encounter pair is much slower than the rate at which i t forms products, that is k b«k^, then k 2 becomes equal;to k^, k k, P d k„ - = k, . (1.6) 2 k d . P In this limit, the rate of the reaction is determined by the rate at which I I the species diffuse together through the medium. This is called the dif-ii fusion controlled limit , and the reaction is diffusion controlled. Typi-9 -1 -1 -1 cally, these reactions have k„ > 10 M s or E of about 17 kJ mole in 2 — a water. For example, radical and atom recombination reactions are often diffusion controlled because the combination of species with unpaired electrons involves very l i t t l e activation energy. The other limit arises when a substantial amount of activation energy is involved in the reaction n step, and then k^ << k^. This gives rise to the activation controlled I I limit and the overall rate constant becomes: -11-(1.7) where K is the equilibrium constant for the A, B and {A...B} species. In this case, the rate depends on the accumulation of energy in the encounter pair as a result of i t s interaction with the solvent molecules, and on their rate of passage over the reaction energy barrier. In the diffusion controlled limit, the rate constant depends on the interaction distance and the relative diffusion coefficients. In general, i t is given by equation (1.8) [56], k 2 = 2 T T ( 1 0 " 3 )N O ( R A + R b ) (D A + D b ) ; (1.8) where R. and D . are the effective reaction radii and diffusion coefficient l l of the i-th molecule, respectively; and No is Avogadro s number. The dif-fusion constant is normally [54] related to the solvent viscosity (n) and * the molecular radius of the solute (r ) through the Stokes-Einstein equation (1.9), D = (1.9) 6iTnr where k is Boltzmann s constant and T the absolute temperature. By assuming r*=Ri and by substituting equation (1.9) into (1.8), one obtains [57] the -12-well known Smoluchowski equation (1.10), No 1000 \ 3n RA- RB (1.10) It is interesting to note that for two species of comparable size, the Smoluchowski equation becomes independent of molecular size. When the. temperature dependence of viscosity is taken into account, the following equation (1.11) applies [58], No 1000 kT 3d RA- RB exp \ kT (1.11) where E is the activation energy of viscosity. l.C.2. Transition state theory applied to liquid phase reactions. ! l " 11 The idea that an intermediate complex was formed during transition from the i n i t i a l reagents to the final products has been known for some time and i t was f i r s t demonstrated by Bjerrum [59] and Bronsted [60] in 1925. However, their derivations using one dimensional coordinates cannot describe the great variety of kinetic features of reaction between complicated mole-cules. Various other groups [61,62,63] developed this idea further to sug-gest that the chemical reaction be considered as the movement of the reacting I I I I I I system along a potential energy surface or phase space . The transition -13-state theory or the activated complex theory were independently formu-lated in 1935 by Eyring [64] and by Evans and Polanyi [65]. Their treat-ments of chemical reaction rates consider the movement of the configura--tional point over a potential energy surface such that the energy barrier between the i n i t i a l and f i n a l states is minimized. The highest point along this path is referred to as the transition state or activated complex. The potential energy surface of most reactions is three-dimensional, and consists I I I I I I M of two valleys connected by a saddle or a pass on the energy barrier [Figure 1.2a). The transition state is considered to be situated on the saddle point of this free energy surface [66]. The curve in Figure 1.2b illustrates the mapping of the potential energy surface E for the following reaction: A + B > » Products . (1.12) along the reaction coordinate through the saddle point, which corresponds to the activated state. The section of the curve at the apex with an a r b i t r a r i l y selected length A£ corresponds to the activated complex (X^ ). This approach led to considerable simplication of the equations for reaction rates and made i t possible to examine the influence of the reactant structure and the nature of the solvent on the reaction rate. Application of s t a t i s t i c a l mechanics to the activated complex theory [54] gives the following expression for the chemical reaction rate of equation (1.12) as: Figure 1.2: (a) Diagram of the reaction path through the energy barrier. (b) Curve of the potential energy plotted against the reaction coordinate, (after Entelis and Tiger [66]) -15-(1.13) where k is a transmission coefficient allowing for the possibility that not a l l activated complexes lead to products since some may be reflected back to the reactants; Ov^/W^g) is the partition function quotient permitting eval-uation of molecular complexity; and A E q is the difference between the lowest energy levels of A and B, and X*, which corresponds to the activation energy for the reaction. l.C.3. Isotope effects and quantum mechanical tunnelling. The application of the activated complex theory to the study of kinetic isotope effects has been quite satisfactory especially with the substitution of deuterium for hydrogen in reactant molecules. For example, i f a reaction involves cleavage of an H-X bond, i t is frequently found that deuterium sub-stitution reduces the rate by a factor of ten at room temperature [54]. How-ever, such a large kinetic isotope effect is confined to reactions involving hydrogen atoms and is rarely seen for other atoms. Using deuterium and tritium, experimentalists and theoreticians have been studying these mass-dependent effects in solution and gas kinetics for decades. Now, with Mu, a superlight isotope of H atom, one has a factor of nine in sensitivity and one can test severely the theory of absolute reaction rates. One can expect bigger isotope effects and therefore Mu can be used to investigate zero-point energy effects, -16-steric and orientational factors, quantum mechanical tunnelling, reaction mechanisms, and the dependence of translational diffusion on the reagents volume and mass. It should be noted that since the muon has a lifetime of -2.2 ysec, there cannot be any stable Mu-containing compounds formed. This renders such discussions as spectroscopic and thermodynamic effects very d i f f i c u l t and limits our focus on kinetic isotope effects. According to classical mechanics, a particle cannot cross a potential energy barrier i f i t s energy is less than the barrier height. However, quantum mechanics states that a moving particle of atomic or electronic mass behaves as a wave packet and predicts that there is a f i n i t e proba-b i l i t y for the crossing at lower particle energies and therefore can often make an important contribution to the overall reaction process. Mathema-t i c a l l y , for a particle m encountering a barrier of height V with wavefunct-ion i t s motion can be solved via the Schrbdinger equation. The solution for m having energy E penetrating barrier V is a wavefunction of the oscillating form [54] (see Figure 1.3). But once inside the barrier ( X > X Q ) , i|> is non zero and decays according to: *(x) = a-exp{-[2mCV-E/ti2]Jsx} . (1.14) Since ^(x)^0, this means there'is a probability of finding m inside the barrier even when E<V. This phenonmenon in which a particle seemingly pene-trates into the forbidden classical region is termed quantum mechanical tunnelling. The tunnelling probability is a sensitive function of the V Figure 1.3: The dependence of a wave f u n c t i o n on the p o t e n t i a l . .For X < X Q > ^ i s o s c i l l a t o r y . For x>x , i|i decreases e x p o n e n t i a l l y ( a f t e r P i l l i n g [ 3 4 ] ) . -18-barrier height and shape, and of the particle s mass and energy. The impor-tance of tunnelling effects in chemical kinetics were recognized as early as 1933 by Bell [67], particularly for reactions involving a proton or hydrogen -atom transfer. Since then, considerable research has been directed towards the design and execution of experiments that would provide examples of tun-nelling effects in chemical processes. It has been observed that hydrogen atoms tunnel to an observable extent, and they do so more effectively than deuterium or tritium atoms because of their lighter mass [68]. This effect might be much more pronounced with the now much lighter Mu atom. Indeed, there is evidence from the muonium lifetime in neat methanol at low tempera-tures [69], and temperature dependence of Mu reacting with NO^  [70] that this is so. There are three major phenomenological manifestations of quantum mechanical tunnelling, curvature in Arrhenius plots, effect on Arrhenius pre-exponential factors and relative kinetic isotope effects. These can be 1 2 investigated by measuring and comparing rate constants of Mu, H, H, and 3 H with various solutes as a function of temperature. It is for this reason, and for reaction kinetics in general, that the temperature dependence of Mu atom reaction rates with different substrates constitutes an important, fundamental, and interesting study in the f i e l d of muonium chemistry. -19-l.D. Description of thesis content. Since this Ph.D. work is an on-going project from the M.Sc. thesis, i t should be appropriate to describe br i e f l y the M.Sc. projects [71]. The essential results were the temperature dependence measurements of five types of reactions of muonium atoms in aqueous solution between 2 and 92°C by the muonium spin rotation technique [70]. The results showed that for an electron transfer reduction with MnO^  , for an addition to ir-bonds in maleic acid, and for a spin-conversion reaction with N i 2 + , the rate constants were a l l diffus-ion limited. An activation energy for diffusion of 17±2 kJ mole ^ was found, 13 -1 -1 while the A-factors varied somewhat in the range of 10 M s . For these diffusion controlled reactions there was no kinetic isotope effect when com-pared with H atoms, so the diffusion coefficient, even for these very small light species, is mass independent. On the other hand, for the abstraction reaction with formate ions, there was a large kinetic isotope effect and the rate constant was activation controlled. The reaction between muonium and NOg was also studied and tended to exhibit a curved Arrhenius plot even over this short temperature range. Such curvature was consistent with either a contribution from quantum mechanical tunnelling or alternative reaction paths. Other temperature dependence studies of both the muon and muonium atoms in C S 2 , ^2®' a n c* w e r e also reported. In addition, the inter-actions of Mu with 0 2 and with porphyrins in water were both investigated. In the case of O 2 , the rate constant, which could be due to spin-conversion, was found to be 2.4x10^ M *s * and was compared with the diffusion-controlled limit and the corresponding gas phase reaction [32]. For the -20-reaction of muonium with porphyrins, the rate constants with hemin and the protoporphyrin were found to be 2.7x10^ and 6x10** M *s respectively [37]; the reaction mechanism was explained via Mu addition to the conjugated double iond for the protoporphyrin and reduction or partial spin-conversion for the hemin substrate. In this Ph.D. thesis, not only w i l l the muonium reactivity with various solutes be emphasized, but also i t s formation probability in neat solvents w i l l be detailed. In chapter 3, the origin of muonium relaxation in pure water (A ) w i l l be elucidated. In chapter 4, a novel technique, muonium-radical spin rotation, w i l l be introduced. This is applied to study muonium selectivity in benzene-styrene mixtures [73]. In chapter 5, muon and muonium yields in concentrated hydroxide solutions [72] and neat neopentane [35] (liquid § solid) are discussed, while the following chapter gives the kinetics of Mu reacting with deutero-formate and hydroxide ions as a func-tion of temperature [72] and the results compared with previous temperature dependence data. Finally, collaborative studies involving muonium formation in hydrocarbons [33,74], muonium kinetics in aqueous solutions of vinyl monomers, nickel (II) cyclam and cyanides [40,41,93,123], and muonium as a bio-logical probe in micelles and cyclodextrins [36,38,39] are presented in chapter 7. It is hoped that the brief description of theories and techniques in this and the ensuing chapter w i l l give the reader a better enjoyment of the presentation of results in the latter chapters. However, i t is recom-mended that references [4,5,28,75] be read by those wishing to further under-stand the physics of the (muon-muonium-radical) spin rotation techniques in condensed media. -21-CHAPTER 2 THEORY AND EXPERIMENTAL METHOD -22-2.A. Theory of the experimental method. Cosmic radiations that reach the surface of the earth are mainly in the form of highly energetic positive and negative muons. They are produced in the earth s atmosphere by incoming protons via pion (TT°, U , ir ) intermedi-ates. This combination of cosmic bombardment and atmospheric conversion is simulated in meson-facilities. 2.A.I. Muon production. Muons are unstable particles and they are decay products of pions. At 9 TRIUMF, the pions are generated by bombarding a Be target (T2) with 450-500 MeV protons from the cyclotron to give the following nuclear reaction. 9Be + *p » 1 0Be + TT + (2.1) The pion then decays with a lifetime (t) of 26 nsec to give a muon and a muon neutrino. TT + » y + + (2.2) Subsequently, the muon i t s e l f decays (x=2.2 ysec) giving an easily detectable positron, an electron neutrino and muon anti-neutrino. -23-2.A.2. Muon polarization. For the pion decay process (equation 2.2), the pion has zero spin and since a l l in nature have negative he l i c i t y (the particle spin is repre-sented in a left-handed sense to the direction of the particle momentum), this forces the muon to also have negative he l i c i t y according to the conser-vation law of angular momentum. In a fortunate but yet significant sense, I I I I this parity violation of weak interactions ensures and provides the experi-menters with a beam of 100% polarized muons. As i f one was not already blessed with luck, in the y + decay (equation 2.3), again due to the non-conservation of parity implicit in the definite h e l i c i t y of neutrinos, the e + emitted also tends to come off along the direction of u + spin. As a result, the angular distribution of the positrons are anisotropic. In gene-r a l , the number of positrons emitted at an angle 6 relative to the muon spin is given by l+acos0 [76]. Here, a is the so-called asymmetry parameter. It is a function of the emitted e + energy with an average value of 1/3 and a maximum value of 1 (for positrons of maximum energy, 52 MeV [4]). It is this variation of the positron distribution which allows one to observe the evolution of the muon polarization and which consequently gives one a sensi-tive way of measuring Mu reaction probabilities. -24-2.B. Experimental techniques. Due to the nature and uniqueness of this relatively novel technique, a l l the experiments were carried out at the TRIUMF cyclotron (originally coined for TRI-University Meson F a c i l i t y , but now governed by four univer-sities - UA, SFU, UVIC § UBC). This meson f a c i l i t y , located on the U.B.C. campus, was completed in the f a l l of 1974. Its capability of producing two simultaneous beams of protons enabled the f a c i l i t y to include such fields of research as medium-energy nuclear physics and muon chemistry, as well as applied research, including medical radioisotope production, neutron activa-tion analysis, pion cancer therapy, nuclear fuel research and positron emission tomography. Figure 2.1 depicts the TRIUMF site and shows the many available beam line channels, experimental beam ports and f a c i l i t i e s . A l l the experiments in this thesis were conducted on the M-20 muon channel. ii it Figure 2.2 shows the different mode of beam operation. In conventional mode, pions decay in flight to give two types of muons, forward and backward muons (in the pion rest frame, forward y + decays in the pion momentum direc-tion while backward y + decays in the opposite direction). Usually, a back-ward muon beam was selected because of i t s clear separation from beam posi--2 tron contamination. It has a penetration of 2 to 3 g cm and momentum of about 80 MeV/c. It is especially well suited for Mu-radical experiments at high magnetic fi e l d s , reasons for which w i l l be explained later. However, the necessity to study low pressure gases [77], or thin solids [78], or small quantities of liquids held in thin-walled cells [29] requires muons of much lower energy. In 1971, a new type of y + beam (Arizona mode) using muons E X I S T I N G PROPOSED MM SPECTROMETER H - POLAR IZED ION SOURCE ISOTOPE LABORATORY NEUTRON ' ACTIVATION ANALYSIS Figure 2 .1 : Schematic diagram showing the TRIUMF s i t e at U.B.C. -26-Q8 Q9 Figure 2.2: Schematic diagram showing the set-up of quadruple and bending magnets for the M20-muon channel at TRIUMF. It also differ-entiates between the types of muons available by tuning for high or low momentum. -27-from the decay of pions that stop near the surface of the production target (i.e. come to rest in T2 as in Figure 2.2) was investigated by a group from the University of Arizona operating at the Lawrence Berkeley Laboratory 184 cyclotron [79J. This was achieved by tuning for a momentum of less than 39 MeV/c, and the resulting beam gave a significant increase in muon stopping density over that of a conventional beam. At TRIUMF, our rapid interchange of targets and use of thin-walled sample cells necessitated the I I I I tune for this, so called, surface muon beam on the M20-beam channel. The use of either backward or surface muons depends on apparatus limitation, sample geometry and type of experiment. In general, the studies of muonium kinetic in aqueous solution employ surface muons while Mu-radical and hydro-carbon investigations require backward muons. 2.B.I. The ySR technique. When a y + stops in a target perpendicular to a magnetic f i e l d (B), i t s spin (1/2) precesses at i t s Larmor frequency, to = y B , (2.4) y ' y ' v J where y = 13.6 kHz/G. To observe this transverse f i e l d precession pheno-menon, one can simply place a small positron counter in the plane of the y + precession at an angle o>o with respect to the i n i t i a l muon spin direction; so at time t the angle between the y + spin and e + detector when the y + decays w i l l be $o+w t. As a result, the e + detection probability N(t) w i l l be -28-proportional to 1+A^cos (<J>o+u)^ t), where is the experimental muon asymmetry (proportional to the muon polarization) and i t s magnitude depends on detector I I n dimensions, target geometry and beam polarization. This method, the ySR -(muon spin rotation) technique, allows one to monitor the magnitude and time dependence of the muon polarization for a l l muons in diamagnetic states, such as free y + or MuH or MuC£, etc. The f i e l d dependence of the diamagnetic muon n I I precession in CC£^ is illustrated in Figure 2.3. The original raw histogram has the form of equation (2.5), - t / t -X t N(t) = NQ-e y[l+A De cos (a)nt+<f>n)J + BG , (2.5) where N q is a normalization factor, BG is the time-independent background, is the muon lifetime (2.2 ysec), A^ is the experimental diamagnetic asymmetry or amplitude, X^ is the exponential decay rate constant, is the precession frequency, and ^ is the i n i t i a l phase of the diamagnetic muon precession. 2.B.2. The MSR technique. In 1976 [24] i t was found, provided the liquid is relatively pure and oxygen free, that Mu can be observed directly by the MSR (muonium spin rotation) technique. This method is a result of hyperfine interaction bet-ween the magnetic moment of the muon and that of the electron (hyperfine f i e l d = B Q = 1585G) giving rise to two net spin states of F=0 (singlet) or F=l ( t r i p l e t ) . Classically, the observable spin state is that of the t r i p l e t to ^ - 0 . 1 0 • 0 . 3 0 0 . 0 0 . 5 1 . 0 1 .5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 0 . 3 0 i n tt - 0 . 1 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 T I M E IN u S E C ( 1 6 N S E C / B I N ) Figure 2.3: Time histograms showing the f i e l d dependence of the diamagnetic precession s i g n a l at (a) 100G and (b) 1500G f o r target s o l u t i o n of CCJl^. The l i n e i s the computer f i t to the muon components i n equation (2.5). -30-which precesses in weak fields at 1.39 MHz/G, about 103 times faster than the free muon in the same magnetic f i e l d . Experimentally, the positron detection probability in this MSR method now acquires another term of the form 6=<J> +uj^ t with a new empirical amplitude A^. Therefore, one has the Mu precession superposed upon the y + precession, while both are superposed upon the expon-ential decay of the muon. In addition, there might be exponential damping on the Mu oscillatory term due to possible phenomenological depolarization and/or chemical reaction. Here, the positron time distribution takes the form of, -t/x - A t N(t) = NQe y[l +A Dcos(a) Dt+<() D) +A Me cos ( o ^ t - ^ ) ] + BG (2.6) 11 I ! I ! I ! where A^ is the Mu-decay rate constant. The raw and asymmetry MSR histo-grams are displayed in Figure 2.4. It should be noted that the tr i p l e t fre-quency that one observes in the MSR method is degenerate at low fields. At higher fields there is a l i f t i n g of this degeneracy as evident from the Breit Rabi diagram in Figure 2.5. This phenomenon w i l l be investigated in the ensuing chapter. However, explicit time-dependent solutions to the muon polarization in muonium are given in references 4, 5, 28 and 75. -31-7 0 0 0 0 .0 6 0 0 0 0 .0 5 0 0 0 0 . 0 4 0 0 0 0 . 0 h ~? 3 0 0 0 0 . 0 ° 2 0 0 0 0 . 0 1 0 0 0 0 .0 0 . 0 0 . 2 0 0 . 10 E < 0 . 0 0 h - 0 . 1 0 h - 0 . 2 0 1 2 3 T i m e in /usee (8 n s e c / b i n ) Figure 2 .4: "Raw" and "asymmetry" MSR histograms of water. The l i n e s i n pl o t s (a) and (b) represent the computer s best f i t s to equations (2.6) and (2.15), r e s p e c t i v e l y . -32-Figure 2.5: Breit-Rabi diagram for a two-spin-% system (o) e=electron Larmor frequency). At low f i e l d s , coe<u)o, § a )23 a r e degenerate, giving r i s e to the c h a r a c t e r i s t i c Mu-precession in the MSR method. At high f i e l d s , to ><D , 0 ) 1 O § <D,. are observable i n 6 e o' 12 34 the MRSR method. -33-2.B.3. The MRSR technique. When a Mu-atom adds to a double-bonded system such as benzene, Mu + (2.7) H H H H a radical compound w i l l be formed; in this case, a cyclohexadienyl radical, •C^ H^ Mu. At low magnetic fi e l d s , there are many transitions with the muon polarization distributed among them. This complexity delayed the direct observation of Mu-radicals until 1978 [26]. By applying much higher magnetic fields (B>2 kG), the vast amount of radical frequencies collapsed to give just 2 transitions (Figure 2.5, co^ § to^ for tue>cuo). The numerical solution to this problem was solved by Roduner et al. [80]. It was shown that by applying high transverse magnetic fields, the muon hyperfine coupling const-ant (a ) in the radical is given by \ = "12 + "43 ( 2 - 8 ) Figure 2.6 shows (a) the raw histogram, while (b) and (c) give the power spec-trum of the Fast Fourier transform (FFT) in frequency space for styrene at several fields. It can be seen that the diamagnetic muon frequency changes linearly with f i e l d , while A^ remains constant with f i e l d . This decoupling method of applying high magnetic fields to observe Mu-radicals is called the 1 2 5 0 0 . 0 -34-1 0 0 0 0 . 0 !Z 7 5 0 0 . 0 5 0 0 0 . 0 2 5 0 0 . 0 h tn o (a) 0 . 0 L 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 1 . 4 1 .6 1.8 2 . 0 TIME IN uSEC (4 NSEC/BIN) O C L $ O C L 0 25 50 75 100 125 150 175 200 225 250 Frequency (MHz) Figure 2.6: Muonium radical in styrene showing (a) the raw histogram at 1500G, and FFT spectra at (b) 1500G and (c) 2500G. -35-Muonium-radical spin rotation (MRSR) technique. Here, the general form of the histogram is given by -t/x N(t) = NQe y[l +A Dcos(co Dt+o3 D) + p i ( t ) ] + B G (2.9) and -X.t R i(t) = A R ie 1 cos (coit+o>i) (2.10) where R^(t) represents various contributions of the i-th radical amplitude (A^) at the i-th frequency (UK) with relaxation rate These three tech-niques (uSR, MSR, MRSR) are shown as FFT spectra in Figure 2.7. 2.C. Electronic logic and experimental set-up. The basic methods (ySR, MSR, and MRSR), whether employing surface or backward muons, involve nuclear physics counting logistics. It requires starting an ultrafast digitizing clock when a muon enters the target sample and stopping the clock when (and i f ) the muon s decay positron is detected in the positron counters within several muon lifetimes. This measured time interval is stored in a time histogram in a PDP-11/40 mini-computer. The process is repeated until sufficient st a t i s t i c s in a time spectrum, such as that in Figure 2.4(a), are obtained. Typical electronic logistics are shown in Figure 2.8. However, more detailed explanations of the electronic and computer logistics can be found in reference [75]. -36-WATER AT 8G, 90CB, R-CELL, BUBB o CL 5.E-4 4.E-4 3.E-4 2.E-4 1.E-4 0 -1.E-4 1 r-«—muon 7 1 SURFACE • i i MUONS —i 1 : ! -—muonium MSR —-I i _ / \ 1 1_ i i i i i i i 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 O D_ .0032 .0028 .0024 .002 .0016 .0012 8.E-4 4.E-4 0 -4 . E-4 STYRENE AT 500G, 90CM, R-CELL, BUBB t k ^ A l 1 r SURFACE MUONS muonic radical MRSR 0 100 200 300 400 500 600 Frequency (MHz) Figure 2.7: FFT spectra showing precession frequencies of the three techniques (ySR, MSR, MRSR). -37-L1-L2-L3 e(LEFT) COIN. . START STOP CON80-CLOCK interface UNIBUS R1.R2-R3 e(RIGHT) COIN. PDP-11/40 UNIBUS / COMPUTER UBC COMPUTER 8: Simplified schematic diagram showing data acquistion system TRIUMF. -38-2.C.I. Surface muon set-up. The counters and target arrangement for surface muons are shown in 11 Figure 2.9. Incident muons which are collimated to 1% beam by lead bricks, I I I I trigger the thin (TM) counter (-15 mil) sending a y-stop signal to the ' i I I start input of a time digitizing converter (TDC) clock; then decay posi-trons detected either by the left (L1-L2-L3) or the right (R1-R2-R3) coin-i t i i cidence telescopes trigger a signal to the stop input of the TDC clock. The resultant time interval is then routed to the appropriate le f t or right histograms. Each histogram is divided into 2048 bins of 2 ns/bin. The I I I I snout was simply an extension beam pipe used to bring the low energy sur-n face muons as close to the targets as possible. The 2 -carbon absorbers between the e counters served to reduce accidentals and to discriminate against low-energy decay positrons, thereby raising the experimental asym-metry [4,27,75]. The magnetic f i e l d is provided by either a pair of helm-I I I I M holtz coils 24 in diameter or sets of rectangular (20 x40 ) helmholtz coils, both of which can generate f a i r l y homogeneous fields (up to -800G) trans-verse to the muon spin direction. As drawn in Figure 2.9, there are two possible positions for the target samples. The common set-up is for the target to be placed at a 45° angle to the muon beam. Since the 4.1 MeV muons w i l l be stopped within 1 mm in most liquids, the left histogram can be used to measure absolute asymmetry values. Most ySR and room-temperature MSR experiments are carried out in this mode of operation. For reasons which w i l l be explained in section 2.D., the temperature c e l l is placed perpendicular to the incident y + beam. Not shown in the diagram are the -39-Pb WALL Ul Q < S B I CI 4.1 MeV SURFACE MUON Pb WALL \ \ \ Pb - COLLIMETER TM COUNTER S 4> ' /<©/ • TEMP - CELL ui O 2 o U l o I Figure 2.9: Diagram of surface muon apparatus geometry. The magnetic f i e l d is physically out of the paper which is called the transverse f i e l d method. See text for explanation. -40-i t i t many lead-bricks which must be painfully stacked around the counters to prevent contaminations from beam positrons which seriously interfere with the time histograms of decay positrons. 2.C.2. Backward muon set-up. The experimental set-up for backward-muon beam is depicted by Figure 2.10. The difference between this and the surface-muon set-up arises from the higher energy of the arriving backward muons (~29 MeV). For cl a r i t y , the diagram shows only one set of positron detectors. In reality, there + I I i t I I n are usually 3 sets of these e counters giving forward , backward , and I I I I perpendicular histograms. As shown, A is lead shielding, DEG is a remote-ly adjustable water block to degrade the muon energy, B is a set of lead I I i t collimators, C 6 D are the muon start counters; S is the sample, T is a temperature box; H are the magnetic f i e l d coils (up to 4kG) giving a f i e l d out of the paper so that the longitudinally polarized muon precesses in the i t I I plane of the paper; E § K are the positron stop counters. As is evident from the diagram, backward muons can pass through the sample. Therefore, in order to distinguish decay positrons from muons, one must use electronic logistics (vetos) to separate the two particles. For a good event in the i t I I MSR or MRSR histogram, the muon start signal is registered by CD.E while + i t i t the decay e stop signal by E-K-D. This set-up is used for experiments involving Mu-radicals and hydrocarbons. For MRSR experiments, histograms are usually of 3000 bin length with 1 nsec/bin packing. The smaller bin packing increases the frequency range and sensitivity which is necessary for observing high frequencies up to 500 MHz. Figure 2.10: Experimental set-up for backward muon experiments. See text for explanation. -42-2.D. Sample preparation and target-holders. A l l solid chemicals were either of reagent or analytical grade and were used without further purification. However, the water used was always care-ful l y purified and t r i p l y - d i s t i l l e d before using for aqueous concentrations. In the case of hydrocarbons such as neopentane, styrene and benzene, they were a l l purified by shaking with H^ SO^ , washing with a base, drying over-night by zeolite, and d i s t i l l e d at least twice several days before an experi-ment. A l l hydroxide MSR experiments were titrated with potassium hydrogen phthalate to obtain concentration values to within 1%. Analogously, the D C O 2 solutions were obtained by titrating formic-acid-d2 (98 atom %D) with a base of known concentration. For the micelle and cyclodextrin samples, the chemicals were specially obtained from the Strem Chemical Company and fresh-ly prepared just before experiment without further purification. With ySR experiments involving solvent mixtures, the neat solutions were of spectro-grade standard and were used directly from the manufacturers' containers. 2.D.I. Surface muon sample holders. A l l MSR experiments involving kinetics in aqueous solutions and most ySR hydrocarbon projects were carried out with surface muons. The low energy muons can be conveniently stopped in thin walled c e l l s . For most aqueous MSR experiments (except the temperature dependence measurements), the liquid samples were contained in a specially designed plastic c e l l (see Figure 2.11a). The body of the c e l l was made of teflon and was cylindrical, 100 mm 11 11 in diameter and 6 mm deep. Its muon windows consisted of very thin sheets -43-O-RING 1 -mil\ rr i r ^ T 2- mil—MYLAR 3- mir o o z o 8 O H -BEAM o o H 20 O a a. o o H20 PC Figure 2.11: Side view of (a) regular t e f l o n c e l l , and (b) temperature t e f l o n c e l l for surface muon experiments. -44-of mylar (0.076 mm). The windows were held onto the teflon c e l l by rubber pressure rings which was held together with screws and plexiglass plates to avoid possible air diffusion. Tiny holes were d r i l l e d into the teflon body -and sealed with teflon plugs. Tygon tubings were attached to the plugs so the solutions can be deoxygenated with high purity He gas before and during each experimental run. It should be noted that the fraction of muons stopped in the mylar window was less than 5% and therefore did not affect the MSR results to any significant extent. Since the incident surface muons were completely stopped within the f i r s t 0.5 mm of aqueous solution with the c e l l placed at angle of 45° to the beam, the left-histogram gives close to absolute asymmetry values. Most aqueous and non-aqueous uSR experiments were also done with this mode of operation with the exception that not a l l the samples were degassed to remove residual 0 2 [32]. For the temperature dependence measurements, a special temperature-regulated c e l l was designed (Figure 2.11b). Similar to the regular c e l l , the reaction vessel was made of teflon with thin mylar windows (two pieces of mylar, total of 0.076 mm separated by an a i r gap of 5 mm) at the front to let the weakly penetrating muons into the medium and to prevent excessive heat exchange with surrounding elements. For temperature regulation, a copper jacket was pressed against the rear mylar window of the c e l l through which water was flowed from a thermostated bath at selected temperatures from 0 to 96°C (see Figure 2.12). Copper-constantan thermocouples sticking into the solution through sealed teflon plugs continuously monitored the temperature of the solution which was actually being bombarded by muons. -45-Figure 2 . 1 2 : Schematic diagram of the surface muon temperature-dependence experimental set-up. -46-There was a small temperature gradient ( 1°C between front and back) even though the cells were only 6 mm deep and the solution was being continuously stirred by the deoxygenating He gas. Moreover, there was never more than a .5°C difference between the bath and c e l l temperature, so possible errors in the temperature scale in the Arrhenius plots were small. Also, in order to avoid unknown variations in the impurity levels, the same solution was studied at several temperatures whenever this could be done without having to change the solute concentration. In several cases, the bubbling solution was lef t untouched and only the bath temperature varied. Furthermore, the highest temperature experiment was invariably done f i r s t to ensure that any extra impurities which may be leached from the c e l l at the elevated temper-ature would be common to a l l solutions and hence to the background X Q 11 ii ( background muonium relaxation constant). Notice in Figure 2.9 that the temperature targets were placed perpendicular to the muon beam and behind the e + detectors. The reason for this is that in the 45°-mode operation, beam positrons (higher energy than decay e + from y +) are scattered e f f i c -iently by copper and this produces a strong cyclotron signal at 23 MHz. This phenomenon effectively smears out the slower MSR signal (-11 MHz at 8G) in the right histogram. Therefore, the perpendicular geometry was the best site for the cells in the temperature experiments. -47-2.D.2. Backward muon sample holders. Due to the higher energy of backward muons, thicker containers (rela-tive to surface muons) must be used to ensure maximum muon stops in the sample. For hydrocarbons such as neopentane, the samples were deoxygenated by freeze-pump-thaw cycles and then vacuum-sealed in a 70 ml round-bottomed glass bulb (~45 mm diameter). The adjustable water degrader was used to stop muons in the middle of the sample. Temperature control was achieved in a closed styrofoam box (Figure 2.10, symbol T) through which cold He gas was passed at a variable rate. Temperatures ranging from room temperature to -150°C were obtained and determined by a thermocouple in contact with the glass-bulb wall. Throughout the experiment, the temperature was monitored continuously with a mV-chart recorder. The temperature was maintained with-in at least 2°C of the quoted values for a l l MSR hydrocarbon experiments. For the MRSR experiments, some were done in sealed-deoxygenated glass c e l l s . However, most were contained in 50 ml round-bottom flasks connected with teflon stopcocks and deoxygenated by bubbling with pure He gas before and during the experiment. Appropriate prebubblers ensured that evaporation did not significantly alter the compositions of various mixtures. 2.E. Data analysis. In condensed phase studies, the highest muon asymmetry obtained is that of CC£^. Its value is also equal to that found in metals such as aluminum and copper [28]. For this reason, i t s value has been used as a normalizing standard for a l l other solvents. In this thesis, the value of is taken -48-as the observed muon asymmetry in the liquid of interest divided by the asymmetry found under identical conditions in CCl^. P = D (2.11) D A i.e. represents the fraction of incident muons which are observed in diamagnetic states. P^  values are usually obtained by the ySR technique under a transverse f i e l d of 80G. For the case of muonium, P^ (the muonium fraction) is twice the value of the Mu-experimental-amplitude divided by the muon asymmetry in CC£ 4. 2 \ A P = (2.12) M " A CC£ 4 The factor of two arises from the fact that half of the muonium ensemble (singlets) are depolarized by the extremely fast hyperfine frequency (co = 4464 MHz) and remains unobservable in the MSR technique. To obtain o P M, MSR experiments are usually carried out at fields of 8 to 9 G. In most solvents, the sum of P„ and P_ is usually less than 1. The M D M I I difference (1-P^-Pp) i s referred to as the missing fraction , P^, which at present is attributed to either unobservable muonium radical formation or rapid spin depolarization perhaps by intra-spur spin-exchange reactions [25,47,52], For observable muonium radical amplitudes obtained in the MRSR -49-experiments, they are also normalized to CCl^ at fields of 1500 to 3400 G. E A R i P D = J : (2.13) A CCS,. 4 By measuring P^ , P^ § P R for a particular sample using the above three tech-niques, one can obtain the various fractions of diamagnetic, muonium, or radical contributions formed. In muonium kinetics involving aqueous solutes and hydrocarbons, both ySR and MSR experiments are done to obtain P^  and P^ , respectively. Whereas for radical experiments, the MRSR and uSR techniques are used to deduce P R and P^. 2.E.I. Analysis of ySR and MSR spectra. In ySR experiments about 10^ events are required for a surface y + spec-trum, while about 1% times as many are needed for a backward y + spectrum. The need for extra events is due to the lower beam polarization of backward muons. Typical asymmetry values are shown in the following table. TABLE 2.1 Typical asymmetry values for backward and surface muons. Substance Surface y + Backward y + AD *M *D ^ CCfc, 0.33 0 0.20 0 4 H 20 0.20 0.033 0.12 0.020 -50-7 As one can see, in an MSR experiment, one needs many more events, about 10 events per histogram for surface muons. However, other considerations such as number of discernable precessions (depends on decay rate) and background are also factors in determining the amount of data that one should accumu-late. Usually, one must compromise between experimental time (very limited at TRIUMF since there are many users from four different universities) and sufficient s t a t i s t i c s . Representative ySR and MSR histograms are shown in Figure 2.3 and 2.4. The lines are the computer f i t s to equation (2.5) and (2.6) using the linear-least squares f i t t i n g program MINUIT [81] compiled on an IBM 12-megabyte Amdahl 470 V/8 computer. In the asymmetry histograms, as mentioned pre-viously, the BG, N q and the muon decay are removed to display more vividly the amplitude signals. The equations for ySR and MSR asymmetry histograms are given by the following equations. A(t) = A De cos (coDt+<(.D) (2.14) A(t) = ADcos(tl)Dt+<j>D)+AMe cos (wMt-<j>M) (2.15) In ySR experiments, one is usually interested in A^. Occasionally, such as solids, one also abstracts the parameter Xp. For MSR experiments, the para-meters of interest are and XM. It should be noted that equation (2.5) f i t s 6 parameters while equation (2.6) f i t s 8 parameters. By experience, t i a l guesses are given to the computer as the starting points, then the m i -51-MINUIT program f i t s a l l parameters. This process usually takes very l i t t l e CPU time to reach convergence. Typical MSR asymmetry histograms for aqueous solute studies are shown in Figure 2.13. The parameter X^  varies with the addition of a reactive solute S at concentration [S] in accordance with equation (2.16), XM = XD + yS] , (2.16) where XQ is the background muonium relaxation constant found in pure water (or other media such as neat hydrocarbon solvents). The linear relationship of X^  and [s] is illustrated in Figure 2.14. It is worth emphasizing that never more than one muon at a time is in the solution, so (X-X ) represents a pseudo first-order rate constant as in equation (2.16) and a plot of X versus [s] gives the slope as k^, which then represents the bimolecular rate constant for reaction of Mu with substrate S. Furthermore, the time scale over which the relaxation was observed (0.2-20 ysec) is much later than the slowing down process of the muon, so k^ is indeed a thermal rate constant. It must be noted that solute concentrations for the experiments were select-ed so that the value of X^  f a l l s within the working range of ~5xl0^ to 5 - 1 2 -2x10 s where the x -minimization of data points is s t a t i s t i c a l l y r e l i -able . For temperature dependence measurements, the extracted muonium rate constant (k M) changes with temperature. This can be expressed by the fami-l i a r Arrhenius equation, 0 . 2 0 - 0 . 3 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 T I M E I N M S E C ( 8 N S E C / B I N ) 2 . 5 3 . 0 3 . 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 - 0 . 3 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 T I M E I N L I S E C M 6 N S E C / B I N ) i on I 3.5 Figure 2.13: Typical MSR asymmetry histograms showing the decay of muonium signal at 9 gauss f o r 5 x l 0 - 5 M KMnO. at (a) 3°C, (b) 58°C, and for (c) 3.5><10"4 M NaNO , (d) 1.4><10~3 M ' 2 NaNO^, both at 1°C. The l i n e s drawn are the computer s best x - f i t of the data points to equation (2.6). -53-°= HEMIN. * = PROTOPORPHYRIN i r 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 PORPHYRIN CONCENTRATION (XIO^) gure 2.14: Plot of observed Mu-decay constants (X M) against solute concentrations (hemin and protoporphyrin) at room temperature to obtain k^. -54-1^ = A.exp(-Ea/RT) . (2.17) A plot o£ In k M against 1/T yields the slope as the activation energy (E &/ R) and the intercept as the pre-exponential factor (A). Again, these parameters are obtained via the MINUIT f i t t i n g of the data points assuming a linear relationship between In k„ and 1/T. r M In general, the s t a t i s t i c a l errors from the computer f i t s were less than the variations found from one experiment to another due to unknown random errors arising from small changes in c e l l positions, f i e l d inhomogeneities, impurity levels, solute concentrations, etc. Sometimes they were less, even, than the difference in values between lef t and right detectors. Realistic it ti probable errors in k^ values may be about 20%. Indeed, the values of k^ in this thesis are always within -20% of previously published values where these were available from the work of other groups from TRIUMF [29] or SIN [25]. 2.E.2. Analysis of MRSR spectra. Due to the many frequencies and their small amplitudes in MRSR experi-ments, at least 10-20 million events are required to discern a l l signals. For samples containing just one Mu-radical, the spectra consist of the diamag-netic frequency and a pair of frequencies whose sum corresponds to the A^ value of the Mu-radical. Figure 2.15 displays Mu-radicals in neat styrene and benzene solutions. The formulation of the raw histogram is given by equation (2.9). As the number of frequencies increases, MINUIT-analysis of expression (2.9) in time-space becomes increasingly d i f f i c u l t . Therefore, O Q. Figure 2.15: FFT diagrams showing Mu-radical formation in neat styrene and neat benzene at 3400G. -56-i t is appropriate to apply Fast Fourier Transform (FFT) techniques to the raw histogram and analyze in frequency space [80,82]. However, in order to compare experimentally observable amplitudes, i t is necessary to apply cor-rections (details of which are given in Appendix I) for the reduction in signal caused by the f i n i t e response time of the detection system ( 1.3 ns) and for limitations imposed by the small number of bins per cycle (especially for high frequencies). In this thesis, the parameters of interest are the amplitudes (AR^) and A^. Normalization (after corrections) of the amplitudes was achieved by comparison with the diamagnetic asymmetry in CC£^ at 80G as defined by equation (2.13). Due to the many corrections needed for the MRSR experiments, P D values are not expected to be better than ±20%. However, K i even considering the many corrections i t is encouraging that P R s in this thesis agree within 10% for similar samples performed by groups from SIN [82] and CERN (Synchro-cyclotron in Geneva, Switzerland) [83]. There was no attempt made to interpret the i n i t i a l <JK or A^, both of which are d i f f i c u l t to analyze precisely. The former parameter depends c r i t i c a l l y on an accurate determination of the experimental zero time, while the latter varies in a non-systematic manner between 0.5 and 3 ys The values of <J>. and A. for Mu-radicals w i l l not be discussed further in this l I thesis. -57-CHAPTER 3 ORIGIN OF X IN WATER: o A MAGNETIC FIELD DEPENDENCE STUDY -58-Since the discovery of Mu i n water by P e r c i v a l et a l . [24], X Q values of about 0.25x10^ s * have been obtained i n MSR experiments by groups from SIN and TRIUMF. There are many possible o r i g i n s of t h i s observed Mu background r e l a x a t i o n constant i n water. (i) It may a r i s e from reaction of Mu with impurities (such as r e s i d u a l 0^ or organic i m p u r i t i e s ) . This was deemed u n l i k e l y due to adequate deoxygenation with helium gas during experiment and the many ca r e f u l d i s t i l l a t i o n s days before beam time. Also, the small E & (zero or at most 0.6 kcal mole *) obtained from the temperature dependence measurement of X q ruled out these chemical reactions since any such reactions would give a much higher E [ 7 l ] , ( i i ) Another possible o r i g i n of X q involves intra-spur reactions of Mu with species produced at the end of y + track. This was eliminated e a r l i e r [29] based on arguments that such intraspur processes would not follow f i r s t order k i n e t i c s [84] nor l a s t longer than 10 ^ sec [85]. It was further indicated that the r a d i a t i o n dose l e v e l s (mainly from positrons accompanying the muons) of less 2 than 10 rad produced within the experimental timescale could not p o s s i b l y account f o r s u f f i c i e n t production of H 20 2 c o n c e n t r a t ^ o n t o § i v e diamagnetic products v i a reaction of Mu with ^2^2' Mu + H 20 2 > MuOH + OH. (3.1) ( i i i ) The only other possible reaction of Mu i s that with solvent water molecules. This w i l l be commented upon l a t e r i n section 3.A.2. Two further possible o r i g i n s of X q are due to (iv) magnetic f i e l d inhomogeneities, and (v) c o l l i s i o n a l broadening, or Mu-frequency beating. Since (i) and ( i i ) -59-have been discounted, one can write the various possible o r i g i n s to X Q by the following expression, Xo = Xo + A A H + AFB ( 3 ' 2 ) where X ' , A . . . and X^- represent the contributions to X from solvent A H FB r o reaction, f i e l d inhomogeneities and frequency beating, r e s p e c t i v e l y . Contribution v i a magnetic f i e l d inhomogeneities can be eit h e r eliminated experimentally by using a combination of high-quality magnets and correction f i e l d c o i l s , or corrected-for a f t e r measurement by ca r e f u l mapping of AH around the target sample area. On the other hand, the contribution to X Q v i a Mu frequency beating (Mu-FB) can be measured d i r e c t l y by MSR experiments at several low f i e l d s and then f i t t e d - o u t by appropriate MSR expressions. It i s t h i s so c a l l e d "two-frequency"-MSR technique that one w i l l be using to investigate X, ,- i n t h i s chapter. F i r s t of a l l , what i s t h i s Mu-FB phenomenon? At very low magnetic f i e l d s (B<12G), the c l a s s i c a l Mu precession frequency i s characterized by the two degenerate t r a n s i t i o n s , w23' s ^ o w n ^y t * i e r e g i ° n °f w e < < a 3 0 ^ n Figure 2.5. However, at B>15G, § u23 a r e n o l ° n £ e r degenerate. Therefore, i t i s now inappropriate to describe the Mu-amplitude signal by the degenerate (no beating) expression, equation (2.15). Notice that X ^ i n equations (2.6) and (2.15) i s ref e r r e d to as X i n neat solvents. In order to account f o r o t h i s non-degenerate Mu phenomenon, one uses the f u l l expression describing both amplitudes a r i s i n g from GJ.^ a n c * w23 L"86]. -60-A - A t Apg(t) = -f e 0 {(l+6)cos(co 1 2t-<t, M) + (l-6)cos(co 2 3t-o> M)} (3.3) where, 6 = x ( l + x 2 ) > 2 (3.4) Here, x i s defined as 2co+/coo (io+ = hi |cog |±|to | ]) . In equation (3.3), <o12 and co2  are represented by: co12 = co_ - , (3.5a) and u>23 = to_ + fi ; (3.5b) where, ft = ^ [ ( l + x 2 ) 3 2 - 1] ; (3.6) and to i s the c l a s s i c a l Mu-precession frequency at very low f i e l d s ( B < < B Q ) . It should be mentioned that equation (3.3) i s v a l i d up to about 150G [4,75] and the difference between to23 and to12 gives 2ft. To properly analyze non-degenerate MSR histograms, one must then use the following expression, -t/T N(t) = N Qe y [1 + ApCosGopt +. ^ ) + A p B ( t ) ] + BG (3.7) As one can now see, the contribution of A„_ to A i n equation (3.2) can be rt) O computer-analyzed by using equation (3.7). This Mu-FB phenomenon can be seen c l e a r l y be measuring Mu i n quartz at 17G (Figure 3.1). Therefore, by measuring Mu as a function of low magnetic f i e l d s , one can then extract Apg from A q by applying t h i s two-frequency expression to MSR histograms. Counts N(t) 3 CD CO CD O CO co co o \ a; CO OJ M o cn M o O o o o o o i o O o o o o o o o O o o o o o M b b b b b b b o Asymmet ry o o o b o p o o O n=r Cn r cn L NO O CD J D C CO 13 o Cn O N cn O Cn cn O Power o o i Figure 3.1: MSR measurement of Mu in quartz at 17G. (a) FFT spectrum showing co 1 2 and i o 2 3 around 24 MHz, the much lower frequency belongs to the diamagnetic s i g n a l . Corresponding: (b) asymmetry and (c) time histograms showing the beating e f f e c t of u * ^ and t o ^ i n time-space. The l i n e i s computer drawn by f i t t i n g the data points with equation(3.7). -62-3.A.I. Results. The magnetic f i e l d dependence of Mu i n water was investigated at three low f i e l d s (~4, 6, § 9G) using the surface muon-MSR technique. The regular t e f l o n c e l l containing the pure water sample was continuously bubbled with He gas and l e f t untouched while the f i e l d was varied from one experiment to another. In order to adequately observe the beating e f f e c t of Mu at low f i e l d s , the gate was set to 8 ysec and each experiment was allowed to c o l l e c t at l e a s t 8 m i l l i o n events. T y p i c a l MSR histograms at low f i e l d s are shown i n Figure 3.2. The data was analyzed over the f u l l 8 ysec spectrum with equation (2.6) and equation (3.7). The x values ( q u a l i t y of f i t to data points, best f i t i s when x =1) f o r a l l f i t s were about 1.05. A l l parameters in equation (2.6) and equation (3.7) remain the same f o r both f i t s with the exception that X q decreases and remains constant for the non-degenerate analysis r e l a t i v e to the degenerate a n a l y s i s . The r e s u l t s are given i n Table 3.1 and i l l u s t r a t e d i n Figure 3.3. It can be seen c l e a r l y from Figure 3.3 that i f one does not consider Mu-degeneracy f o r slow Mu relaxations, the Mu-FB e f f e c t appears as an extra contribution to X q and the e f f e c t increases f o r p r o g r e s s i v e l y higher f i e l d s . As shown i n Table 3.1, the Mu-FB contribution to A q i s simply the d i f f e r e n c e between the two f i t t e d values (Apg). It i s i n t e r e s t i n g to note that A P B seems to increase r a p i d l y with f i e l d and suggests a power dependence of about 4. In addition, the magnetic f i e l d around the target sample area was mapped c a r e f u l l y with a Hall-probe. The measurement indica t e d an average f i e l d inhomogeneity (AH) of about lOmG within a 10 cm cube at the center of the c o i l s . 0.20 -63-OJ E E V) < 0.10 0 .00 - 0 . 1 0 cu E E V) < E E < 0 1 2 3 4 5 6 7 T ime in fj.sec (32 n s e c / b i n ) Figure 3.2: . MSR histograms of Mu i n water f i t t e d with equation (3.7) at (a) 3.85G, (b) 6.44G, and (c) 9.38G. -64-TABLE 3.1 A values obtained by f i t t i n g with and without consideration of Mu-degeneracy o at low magnetic f i e l d s . A /10 6 s " 1 Xj™6 s_1 B/gauss o FB  Degenerate-fit Non-degenerate-fit 3.85 0.109±0.055 0.111±0.050 0.002 6.44 0.136±0.063 0.109±0.037 0.027 9.38 0.230±0.051 0.099±0.069 0.131 A q i s obtained by f i t t i n g the XQ is* obtained by f i t t i n g the ApB i s defined as the absolute and non-degenerate values. data with equation (2.6). data with equation (3.7). difference between the degenerate 0 . 3 0 0 0 . 2 5 0 0 . 2 0 0 0 . 1 5 0 0 . 1 0 0 0 . 0 5 0 0 . 0 0 0 0 . 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 7 . 0 8 . 0 9 . 0 1 0 . 0 Magnetic field/gauss ure 3.3: Magnetic f i e l d dependence of X Q for water. ( a ) represents the value obtained by the degenerate expression, equation (2.6), while ( O ) was by the non-degenerate equation (3.7). -66-3.A.2. Discussion. It is obvious that the Mu-FB effect contributes to A . In the higher f i e l d range (B=9G), Apg comprises almost 60% of the A q value. Therefore by subtracting this phenomenon, A Q C = X , + A ^ ) becomes 0.11x10^ s 1 . A similar non-degenerate expression to equation (3.7) was used to f i t low f i e l d MSR histograms by Percival et a l . [87]; however, the authors did not find any difference for A q between the degenerate and non-degenerate f i t s . Perhaps their i n a b i l i t y to see the A „ D contributions was due to combinations of small r B data gate Q<4 ysec), insufficient s t a t i s t i c s , and low asymmetry of backward muons. In order for the computer to properly f i t (or "see") the beating effect, one must collect over long gate times (at least >8 ysec), have high Mu-asymmetries, and certainly a vast amount of data points, especially for the longer time range. 4 It was interesting to note that A C D varies approximately as B . This A-fourth-power dependence on the value of the external f i e l d was qualitatively predicted by Nagamine et a l . [88] from the University of Tokyo Meson Science Laboratory (UT-MSL) at KEK (National Institute of High Energy Physics) in Japan. However, the exact analytical dependence of A p B upon the magnetic f i e l d is not well known. At present, the gas-chemistry group at TRIUMF has concluded that the functional form of the dependence is neither exponential nor Gaussian and they are now performing Monte-Carlo simulations to extract the exact ApB dependence upon B. In any case, from this chapter's result, i t is clear that A p B has a strong dependence upon B and this is demonstrated especially at relatively high fields (-10G). -67-The f i e l d inhomogeneity AH was measured around the target sample area and an average value of about 10 mG was obtained. Since AH causes dephasing of the coherent Mu precession with a rate of Y^AH [88], a corresponding "dephasing rate of 0.09x10^ s * w i l l be expected f o r the AH contribution to A Q ( i . e . . Thus, the present r e s u l t a f t e r subtraction of A p g and A ^ from A q gives a lower l i m i t value f o r equation (3.2) of A = 0.02 x 10 6 s" 1 . (3.8) o Within s t a t i s t i c a l and systematic e r r o r s , t h i s value of A q can be probably considered as zero. This suggests that A q i n water i s simply an experimental a r t i f a c t due to a combination of magnetic f i e l d inhomogeneities and the physical phenomenon of Mu-frequency beating. Let's consider that A q i s non-zero and has a value corresponding to equation (3.8). I f t h i s i s so, t h i s number w i l l be the contribution A ' due to the reaction o f Mu with water solvent molecules. The fol l o w i n g are possible reactions giving r i s e to diamagnetic species such as MuOH, MuH or MuO [29]. Mu + H 20 > H + MuOH (3.9a) Mu + H 20 > OH + MuH (3.9b) Mu + H 20 > H 2 + MuO (3.9c) These reactions do not involve d i f f u s i o n of the reacting species since the -68-water molecules are already present at a concentration of 55M! This places an upper limit on k^ (reaction rate constant of Mu with H20 molecules). 2 -1-1 Using XQ = k M[H 20], one obtains k M <_ 3.6x10 M s . Since this is an upper limit (i.e. AQ can be smaller), i t is not inconsistent with the corresponding k^ of about 10 ^ M *s * [89] for the reaction, H + H20 H 2 + OH . (3.10) Considering a l l the errors and d i f f i c u l t i e s involved in the MSR measurements of small Mu-relaxation rates, this indicates that Mu does not react with solvent water molecules. The fact that A q in water is of such a small value is much blessing for the Mu-chemist studying kinetics of various solutes in aqueous solutions. If AQ was greater than 1 ys \ the range of concentrations (see Figure 2.14) that one could investigate would be very much smaller and place tremendous uncertainty in the evaluation of k^. However, i t should be emphasized (contrary to Nagamine et a l . [88]) that the A q value one uses for extraction of k^ in equation (2.16) should be the value obtained by each group f i t t i n g with equation (2.6). Although one can extract Apg from A q by f i t t i n g with equation (3.7), this can not be done for higher A*s (similar to the case of slower relaxation at short data gate). Since A„ D is the same at constant magnetic f i e l d , this w i l l have to be subtracted out from concentration-A's >_ 1 sec This is equivalent to f i t t i n g with the degenerate expression, equation (2.6), for a l l concentrations including pure H20. This is further illustrated in Figure 3.4. In other words, k M values for various solutes in -69-, > [S] Figure 3.4: Diagram of expected X dependence vs. [s] f i t t e d with (a) equation (2.6) and (b) equation (3.7). Both methods should give the same k^. However, equation (2.6) i s a much simpler expression to use for data analysis i n the MSR technique. -70-aqueous s o l u t i o n should be obtained by using X o ' s and A's f i t t e d with equation (2.6) f o r one p a r t i c u l a r set of experiments. It should now be of p a r t i c u l a r i n t e r e s t to use t h i s two-frequency "MSR technique to investigate XQ of various solvents ( i . e . neopentane, MeOH, etc.) to see i f Mu r e a l l y does react with these compounds. However, MSR experiments with these solvents are extremely d i f f i c u l t due to impurity problems. Non-reproducibility of X q with d i f f e r e n t samples of the same hydrocarbon delays such attempts with neat solvents. However, the i n v e s t i g a t i o n of Mu i n hydrocarbons w i l l be further elaborated i n l a t e r chapters. -71-CHAPTER 4 MUONIUM RADICALS IN BENZENE, STYRENE, AND THEIR MIXTURES -72-As was explained i n section 2.B.3., muonium-containing free r a d i c a l s can be observed i n unsaturated molecules by applying high-enough transverse magnetic f i e l d s . Since the i n i t i a l d i r e c t discovery of Mu-radicals i n 2,3-dimethyl-2-butene [26], more than 80 of these species have been observed by groups from SIN, CERN, and TRIUMF. The MRSR technique can be used to study k i n e t i c isotope e f f e c t s or i n i t i a t i o n k i n e t i c s , s e l e c t i v i t y of Mu-reactions based on i n t e r - and intra-molecular exchanges, e f f e c t s of group su b s t i t u t i o n s , and formation of Mu-radicals and Mu i t s e l f . In pure styrene, i t was shown [41] that only one Mu-radical i s observed i n a solu t i o n of neat styrene -9 -5 (C£HrCH=CH-) on the MRSR timescale (between 10 - 10 sec) . S i m i l a r l y , only one p a i r of r a d i c a l frequency l i n e s i s observable f o r pure benzene [41,90]. In the case of styrene, there are s i x possible r a d i c a l s that can be formed by the Mu addition to the double bonds of the v i n y l or phenyl groups, yet only one r a d i c a l i s observed. Is t h i s s e l e c t i v i t y a r e s u l t of intra-molecular rearrangement to the most stable r a d i c a l following random primary attack, or i s i t a matter of k i n e t i c competition i n forming the i n i t i a l r a d i c a l ? Of course, i n the case of benzene, a l l p o s i t i o n s of Mu-attack are equivalent, therefore only one Mu-radical frequency p a i r i s pos s i b l e . The r a d i c a l frequencies f o r separate solutions of neat styrene and neat benzene at 3400G are shown as FFT spectra i n Figure 2.15. Also, the f i e l d dependence of the diamagnetic and r a d i c a l frequencies are i l l u s t r a t e d i n Figure 2.6. In ESR studies, H-atom addition to the phenyl group of polystyrene gives a primary s p l i t t i n g of aD=45G [91] which i s translated into -73-A =126MHz. On the other hand, i t has been calculated [92] that A f o r an P P ESR r a d i c a l observed i n the backbone of polystyrene should give a value of 56 MHz. To c a l c u l a t e the equivalent MRSR-A (A') from A obtained i n styrene y y y (213.4±0.2 MHz) [41], the difference between the magnetic moments of the proton and muon i s considered; thus A = A (-£•) . (4.1) y ii u y This gives a value of 67.0 MHz f o r A' i n the styrene r a d i c a l (see Table 1.1 y f o r r a t i o o f p to p ). By comparison to the ESR values, i t i s cl e a r that the observable r a d i c a l i n styrene belongs to the Mu addition to the v i n y l bond of styrene. For example, H H H ^ ^H I I Mu + _C = C > M u — C — C • (4.2) H H (o) Therefore, there i s no net i n s e r t i o n of Mu into the phenyl r i n g , yet benzene i t s e l f r e a d i l y forms cyclohexadienyl r a d i c a l s as shown by equation (2.7.) (A^ = 514.6±0.2 MHz) [26,93]. S i m i l a r l y , there are three r a d i c a l s (o-, m-$ p- with A^ - 490-512 MHz) formed i n comparable y i e l d s i n both toluene and ethylbenzene [82,83]. At present, there are four models of Mu-radical formation [94]: -74-* Mu + S > radical (I) Mu + S » radical (II) u+ + S > [S +] > radical (III) + e" + S > S" y > radical (IV) These models diff e r in the nature of the radical precursor. In models I and II, the precursor is assumed to be Mu, with model I being hot and model II thermal [22,95]. However, for models III and IV, the precursor is diamagnetic, be i t [Sy +] in (III) or y + in (IV) [96]. As one can see, the models of Mu-radical formation are just as tentative and even more variable than that of Mu formation i t s e l f . Therefore in this chapter, i t is appropriate to investigate mixtures of benzene and styrene in an effort to seek answers to questions about intra- and inter-molecular exchanges, and possibly about "hot" or "thermal" Mu attacks. It should be mentioned that since the comple-tion of this project [73], a paper reporting similar experiments (from CERN) has appeared [96], However, the results only agree moderately and the discussions differs substantially. 4.A.I. Results. Backward muon set-ups were used to obtain MRSR spectra of styrene, benzene, and their mixtures. A magnetic f i e l d of 3.4 kG was chosen for a l l experiments. Twenty million events were collected for at least one of the three spectra. Already spectrograde samples were r e d i s t i l l e d just before experiments and were contained in glass c e l l s . Before and during the data -75-taking, the samples were bubbled with high purity helium. To ensure evapor-ation did not significantly alter the compositions, appropriate bubblers were used to saturate the He gas before entering the sample solutions. The value of A , A' and A for styrene and benzene are listed in Table y y p 4.1. Figure 4.1 shows the FFT spectra of the MRSR experiment at 3.4kG for the 80% benzene/20% styrene mixture. It consists of the diamagnetic peak (D) and two pairs of radical peaks (S^ § for styrene, § for benzene) whose frequencies are given in Table 4.2. These are the same as found separately in pure styrene or benzene, and their sums correspond to those in Table 4.1 and other published hyperfine coupling constants [26,4l]. In order to compare amplitudes (as detailed in Appendix I), i t is necessary to apply corrections for the reduction caused by the f i n i t e response time of the detection system (-1.3 nsec) and for limitations imposed by the small number of bins per cycle after analysis in Fourier space (section 2.E.2.). The correction factors used for the four radical frequencies are given in Table 4.2. After corrections, the amplitudes were normalized against the diamagnetic asymmetry in CCit^ at 80G. values of 0.17 and 0.15 at 80G were found for pure benzene and styrene respectively, in good agreement with previous results [41,96], No free muonium atoms were found in any of these systems (i.e. P^=0). Figure 4.2 shows how the corrected, normalized radical yield of the cyclo-hexadienyl Mu-radical (RD) and the phenylethyl Mu-radical (Rg) vary with the volume-fraction of the mixtures. When plotted against mole- or mass-fraction the plots are similar. However, i t is more appropriate to use volume-fraction since i t is probably the best way to represent the -76-T A B L E 4.1 Hyperfine coupling constants of styrene and benzene Mu-R H-R Monomer Radical • A /MHz A'/MHz A /MHz Styrene C6H5CHCH2Mu 213.4 67.0 56.0 Benzene C,H,Mu 514.6 161.6 133.7 f 6 D + Taken from reference [97], 0> o CL 1.E-4 7.5E-5 5.E-5 h 2.5E-5 0 B H i 0 100 200 300 Frequency (MHz) 400 500 i Figure 4.1: FFT spectrum of 20% styrene by volume in benzene obtained from the 3.4kG MRSR histogram. Peak D represents the diamagnetic signal (46 MHz), S^ and S 2 (sum = 215 MHz) represent the Mu-radical from styrene, Bj and B^ (sum = 514 MHz) represent the Mu-radical from benzene. (power i s proportional to the amplitude squared) -78-TABLE 4.2 Radical frequencies at 3.4 kG i n styrene (S^ and S^) and benzene (B^ and B 2) and correction factors needed to c a l c u l a t e the amplitudes. Column X gives the observable f r a c t i o n due to the timing r e s o l u t i o n of the detection system, and Y i s the observable f r a c t i o n due to the packing f r a c t i o n (or number of bins per cycle) used in the a n a l y s i s . Radical Frequency/MHz X Y 60 0.97 0.99 154 0.82 0.96 205 0.71 0.94 310 0.46 0.86 10 20 30 40 50 60 70 80 90 100 Volume % of Benzene in Styrene Figure 4.2: Plot of normalized, corrected r a d i c a l y i e l d s against the volume-percent of the benzene/styrene mixture. Upper curve (o) gives P(R q), lower curve (X) p( R B)« -80-f r a c t i o n of time spent by the r a d i c a l precursor i n contact with benzene or styrene molecules. The t o t a l y i e l d s (P n, P § P ) against volume f r a c t i o n of u K L the mixtures are i l l u s t r a t e d i n Figure 4.3. Notice that i n the p l o t , P^ i s the sum of P(R C) and P(R D). The l i n e s are drawn by eye to show the dependence O D of Pp and P ^ + P Q with composition. 4.A.2. Discussion. It i s c l e a r from the FFT mixture spectra (Figure 4.1) that the two r a d i c a l frequency p a i r s are the same as those i n the separate neat l i q u i d s (compare Table 4.1 and Table 4.2). Since both Mu-radicals can be observed -9 -6 i n each mixture within the time window of -10 to 3x10 sec, i t implies that there i s no f a s t inter-molecular conversion during t h i s time scale. The f a c t that they can be seen at a l l i n the MRSR technique places an upper _9 l i m i t of Mu-radical formation of <<10 s. In any case, the observable time-scale, i n c l u d i n g the f a c t that the styrene r a d i c a l s i g n a l i s f a i r l y long l i v e d (X - 1 ys~^), means that the cyclohexadienyl r a d i c a l does not i n i t i a t e polymerization of styrene, so i t s rate constant (k^) f o r i n i t i a t i n g styrene 4 -1 -1 polymerization must be less than -10 M s The r a d i c a l y i e l d p l o t i n Figure 4.2 indicates some sort of k i n e t i c competition, since on a volume basis there i s not an equal p r o b a b i l i t y o f forming Rg and Rg. Let's consider the following reactions: k R X + B - > P(R D) (4.3a) k S and X + S 2 * P(R g) ; (4.3b) where X i s the precursor (be i t Mu, hot or thermal, or a y + i o n ) ; then, -81-100.0 c q D N _ D O C CD U i _ CD C L 75.0 50.0 25.0 0.0 0 M i s s i ng /1_ R a d i c a l s u r -a $ p ft • -n D i a m a g n e t i c 25 50 75 100 V o l u m e % of B e n z e n e in S t y r e n e Figure 4.3: Plot of t o t a l y i e l d s (Pp, P R § P L) against volume f r a c t i o n of the benzene-styrene mixtures. Note that P D i s the sum of P(R„) K S and P(Rg). The l i n e i s drawn by eye to show the dependence of P D and P R with composition. (•) P n alone, ( A ) P n+P D. -82-providing kg and kg remains invariant over the f u l l composition range, the inverse f r a c t i o n a l y i e l d s are given by: 1/P(R B) = l/P°(Rg) {1 + k s[S]/k B[B]} (4.4a) and 1/P(R S) = 1/P°(R S) (1 + k B[B]/k s[s]} , (4.4b) where P°(R ) and P°(R„) are the f r a c t i o n s of Mu adding to benzene and styrene, B 3 r e s p e c t i v e l y . Figure 4.4 displays the l i n e a r dependence of equation (4.4). This strongly corroborates the occurrence of a d i r e c t competition between benzene and styrene f o r a common precursor. Both l i n e s give a r a t i o of kc/k D = 8±1. This r a t i o i s more than twice the value obtained by Cox et a l . O D [96]. Their kg/kg value of (3.5±0.15) was determined by assuming a constant precursor y i e l d of 0.35 ( i . e . P°(R ) = P°(R„)). Indeed, such an assumption B " can p o s s i b l y cause a poor f i t to t h e i r data. However, considering the many corrections that are needed i n analyzing MRSR spectra and the f a c t that both groups (CERN and TRIUMF) used d i f f e r e n t sets of apparatus and e l e c t r o n i c s , there i s probably as good a c o r r e l a t i o n between the two r a t i o s as could be expected at t h i s stage. It can be seen from Figure 4.3 that a large missing f r a c t i o n , P^, i s present i n the neat solutions and t h e i r mixtures. Since there were no free Mu-atoms observable at low f i e l d s , t h i s suggests that P^ does not stem from Pj^. However, no corrections have been made here to the r a d i c a l y i e l d s f o r any decay of the r a d i c a l s during or p r i o r to the observation window; nor f o r any possible dephasing by p r i o r precession of precursor X i f the absolute [ S t y ] / [ B z ] 0.0 40.0 30.0 cn C L 20.0 10.0 h 0.0 0.0 0.1 5.0 0.2 0.3 0.4 10.0 [ B z ] / [ S t y ] 0.5 0.6 15.0 10.0 5.0 TI CO H 0.0 15.0 20.0 25.0 30.0 00 Figure 4 .4 : Kinetic competition p l o t s : upper curve ( X ) represents l/P(Rg) versus [s]/[B] and lower curve ( o ) represents 1/P(R„) versus [B]/[S]. -84-rate constants kg and kg are not high enough [95] ( i . e . , see chapter 5, OH ), nor f o r any d e p o l a r i z a t i o n of the " s i n g l e t " muonium through the hyperfine i n t e r a c t i o n . The observed P could be smaller than the i n i t i a l r a d i c a l K " f r a c t i o n s , and thus contribute to P^, f o r any of these reasons. A s e l e c t i v i t y r a t i o of 8 based on competition i n the primary r a d i c a l -forming step between the d i f f e r e n t solvent molecules can be accounted f o r , as above. However, the i d e n t i t y of precursor X (diamagnetic or Mu) cannot be established unequivocally with such competition experiments above [94], In p r i n c i p l e , measurements of the i n i t i a l Mu-radical phases may be used f o r such a determination. However, Roduner [94], using r e l a t i v e phases, concluded Mu to be the main d i r e c t r a d i c a l precursor i n various solvent mixtures, while Cox et a l . [96] found the opposite r e s u l t ( i . e . X i s diamagnetic) by measuring absolute phases. Such variance of i n t e r p r e t a t i o n may be likened to that of hot and spur models of Mu-formation. The i d e n t i t y of precursor X w i l l probably require more and d i f f e r e n t types of experiment. Regardless of the above controversy, one can discuss the p o s s i b i l i t y of hot or thermal X a d d i t i o n to the mixtures. Comparatively, hot additions of X should be less s e l e c t i v e than that of a thermal process; but i s a s e l e c t i v i t y r a t i o of 8 s u f f i c i e n t l y large to force one to favour a thermal-X addition? I n t r i g u i n g l y , i f X was a thermalized Mu atom, and i f t h i s same kg/kg r a t i o applies to Mu reactions i n d i l u t e aqueous s o l u t i o n , then k^ f o r the benzene 8 -1 -1 9 r e a c t i o n of Mu would be only 1.4x10 M s because k M f o r styrene i s 1.1x10 M - 1 s _ 1 [41]. This rate f o r benzene i s barely f a s t enough to r e t a i n phase coherence i n the muon spin as the r a d i c a l s are formed i n these mixtures. -85-Furthermore, i t would imply an inverse isotope e f f e c t (k^/k^ = 0.16 [99]) which i s h i t h e r t o unknown f o r an addition r e a c t i o n . Due to the above in e x p l i c a b l e isotope e f f e c t , one should devise an -alternative explanation. Roduner [94] has investigated the degree of tr a n s f e r of muon p o l a r i z a t i o n v i a amplitude and phase e f f e c t s and obtained a value of 9 -1 -1 k M = (8.9±0.6)xl0 M s f o r Mu addition to neat benzene. Applying the s e l e c t i v i t y r a t i o of 8, one obtains k,. ~ 7x10^ M '''s * f o r Mu addi t i o n to neat ' M styrene (Table 4.3). Such a high k^ should r e t a i n a l l of i t s muon p o l a r i z a t i o n [95] i f a magnetic f i e l d dependence study i s c a r r i e d out. Such an experiment i s i n the preliminary stages at TRIUMF. Table 4.3 l i s t s the k^, k^ and k^/k,, values f o r reactions i n water, neat solvents, and gases. It i s i n t e r e s t i n g that the s e l e c t i v i t y r a t i o (kg/kg) i s bigger f o r H than f o r Mu by a f a c t o r of two. Such a r e s u l t , with Mu being l e s s s e l e c t i v e than H, i s an agreement with other free r a d i c a l data from SIN [94,100]. This suggests e i t h e r of two p o s s i b i l i t i e s : that the precursor X i s epithermal or that the e f f e c t i s due to the t u n n e l l i n g a b i l i t y of the l i g h t e r Mu atom which reduces i t s s e l e c t i v i t y . These cannot be distinguished at t h i s time. It would be of p a r t i c u l a r i n t e r e s t i f such MRSR mixture experiments and Mu-kinetic studies were c a r r i e d out i n the gas phase. F i n a l l y , there i s also the i n t r i g u i n g question of why only one r a d i c a l structure i s observed i n styrene (but not i n toluene, f o r instance). The fa c t o r of eight favouring styrene over benzene may be consistent with the greater rate of attack at the v i n y l side chain than i n the benzene r i n g . This would account f o r observing only the v i n y l addition aduct for' styrene i f -86-TABLE 4.3 Various k„ and k t I values f o r benzene and styrene M H -1 -1 -1 -1 kM/M s k H/M s 3. b e (in water) (in neat solvent) ( in water) ( i n gas phase) styrene (S) 1.1 x 10 9 [~7 x 1 0 1 0 ] + [~2 x 1 0 9 ] 9 [~8 x 1 0 8 ] 9 k s / k B 1 8 2 16 benzene (B) ~1 x 10 9 [8.9 x 1 0 9 ] # 1 x 1Q 9 5.01 x 10 7 Since k„ i s not a v a i l a b l e f o r styrene, the numbers are i n f e r r e d from H H-atom reac t i n g with ethylene. Taken from reference [94]. Inferred from the experimentally s e l e c t i v i t y r a t i o i n chapter 4. a k^ values of styrene i n water was taken from reference [93], while benzene came from preliminary TRIUMF data. k Values taken from reference [99]. C Values taken from reference [98]. -87-one cannot also resolve the small remaining p o l a r i z a t i o n d i s t r i b u t e d three ways among the ortho, para and meta vinylcyclohexadienyl frequencies. In addition, one cannot observe the C^ .H,-CHMuCH- r a d i c a l e i t h e r . Two explana-tions can be forwarded, e i t h e r i t i s not formed i n i t i a l l y or i t simply disappears too quickly on the ySR time-scale. The former event would suggest an even higher primary s e l e c t i v i t y than the obtained value of 8 since kg would be based on attack on ju s t one carbon s i t e (out of up to 8 i n styrene) compared to any one of 6 equivalent s i t e s i n benzene. In the l a t t e r case -where the observed s e l e c t i v i t y accrues from the disappearance of the other primary r a d i c a l s - one needs to contemplate whether the disappearance arises from intramolecular rearrangement to the most stable r a d i c a l configuration (equation (4.2)), or whether the other r a d i c a l s are l o s t by reaction with reactive species from the muon track or with styrene i t s e l f . From gas phase studies [98], intramolecular r e l a x a t i o n to the most stable r a d i c a l seems appropriately s e l e c t i v e f o r these observations. Therefore, as suggested by Table 4.3, even though the Mu-addition re a c t i o n producing r a d i c a l s i s f a s t (close to d i f f u s i o n c o n t r o l l e d l i m i t f o r styrene), possible intramolecular rearrangements within the MRSR time-scale can induce phase incoherence of the r a d i c a l signals causing low r a d i c a l y i e l d s and the r e s u l t i n g large missing f r a c t i o n s . 4.A.3. Conclusion From these MRSR experiments i n mixtures of benzene and styrene, one can derive the most stable structures. In t h i s case, intramolecular rearrangement gives r i s e to the only observed r a d i c a l i n styrene and the s e l e c t i v i t y r a t i o -88-(k„/k_) of 8 in the mixture. The possibilities of tunnelling and epithermal O D reactions warrant further investigations into the many formation models of Mu-radicals. It is certainly clear that many more experiments are necessary to resolve some of these controversial, and yet most interesting issues, regarding the role of Mu-radical formation in liquids. -89-CHAPTER 5 MUON AND MUONIUM YIELDS -90-Both models, hot and spur, of muonium formation seem to be i n constant consideration by the muon community. Many experiments were performed i n the past several years i n an attempt to c l a r i f y the arguments. Pe r c i v a l et a l . [43], s h o r t l y a f t e r t h e i r discovery of Mu i n water, proposed the spur model of muonium formation. Based upon t h e i r model, they explained the concentra-t i o n and f i e l d dependence of the diamagnetic f r a c t i o n (P^) i n the presence of electron scavengers. Walker [ 5 l ] proposed many arguments against such a spur model based on r a d i a t i o n chemistry e f f e c t s . In t h i s chapter, muon and muonium y i e l d s i n neopentane and concentrated hydroxide solutions w i l l be used to further investigate the proper mechanism of Mu-formation i n condensed media. 5.A. Muonium atoms i n l i q u i d and s o l i d neopentane. Hydrogen atoms are d i f f i c u l t to observe i n l i q u i d hydrocarbons; therefore i t i s p a r t i c u l a r l y useful to use Mu as an isotope of H to study t h i s subject. Long-lived Mu has been detected i n water, and also i n alcohols [42] and saturated hydrocarbons [33], although t h e i r l i f e t i m e s are comparatively reduced. There are three major reasons to study neopentane. ( i ) This compound has a l l of i t s hydrogen atoms present i n methyl groups, thus Mu could be l o n g - l i v e d since abstraction of such H i s slow, ( i i ) The molecule i s almost sp h e r i c a l and therefore i t s quasifree electrons are enormously mobile [ l O l ] , This means that neopentane w i l l have an unusually large free ion y i e l d i n r a d i o l y s i s , despite i t s low d i e l e c t r i c constant. On the basis of the spur model [43], then, one would expect i t s Mu-yield to be dramatically d i f f e r e n t -91-from both water and hexane. ( i i i ) Since neopentane has a convenient melting point and a simple s o l i d structure, i t i s therefore an appropriate and i n t e r e s t i n g medium i n which to study the e f f e c t of phase t r a n s i t i o n s "on the various muon y i e l d s . 4. A.I. Results. Neopentane was c a r e f u l l y p u r i f i e d , deoxygenated, and vacuum sealed i n a 7 70 ml round glass bulb as described i n section 2.D.2. About 10 backward muon events were c o l l e c t e d on each of three positron counters f o r the sample held at four temperatures between 209 and 295 K (melting point = 253 K). P^ and X^ values were obtained at 8G by f i t t i n g the data with equation (2.6), while Pp was obtained at 80G with equation (2.5). The muonium and muon precessions are displayed i n Figure 5.1. Table 5.1 gives the values of A Q (X.,=X since t h i s value i s the observed Mu-decay rate constant i n the neat M o J solvent), Pj^, Pp and P^ i n the l i q u i d and s o l i d phases of neopentane. 5. A.2. Discussion. These r e s u l t s have three major aspects of i n t e r e s t . F i r s t , Table 5.1 provides muon y i e l d s i n a hydrocarbon which has physical properties d i f f e r e n t from both l i n e a r hydrocarbons and saturated weakly polar alcohols and ethers. This i s p a r t i c u l a r l y i n t e r e s t i n g with regard to r a d i a t i o n chemical free ion y i e l d s of these l i q u i d s [102]. As shown i n Table 5.2, neopentane provides a medium due to i t s s p h e r i c i t y and n o n - p o l a r i z a b i l i t y [101] i n which quasifree electrons are exceptionally mobile. The table indicates that f o r solvents -92-45000.0 - 30000.0 g 15000.0 h 0 .0 6000.0 4000.0 h in o 2000.0 f-0.0 1 2 3 TJME JN uSFC 18 NSEC/BIN) F i g u r e 5 . 1 : Raw h i s t o g r a m s ( d o t s ) a n d c o m p u t e r f i t s ( l i n e s ) f o r l i q u i d n e o p e n t a n e a t 2 9 5 K; ( a ) s h o w i n g t h e m u o n i u m p r e c e s s i o n w i t h 8G t r a n s v e r s e m a g n e t i c f i e l d ; a n d (b ) s h o w i n g t h e d i a m a g n e t i c muon p r e c e s s i o n a t 8 0 G . -93-TABLE 5.1 Values of XQ, P^, P D and P^ i n neopentane i n l i q u i d and s o l i d phases. Phase T/K A / l o V 1 0 Liquid 295±3 0.26±0.03 0 .18±0 .02 0 .55±0 .02 0, ,27±0, .03 Liquid 259±2 0.30±0.07 0 .14±0 .02 0 .60±0 .02 0. ,26±0. .03 S o l i d 238±2 0.17±0.06 0 .14±0 .01 0 .59±0 .02 0, .27±0. .03 S o l i d 209±2 0.33±0.13 0 .19±0 .02 0 .61±0 .02 0, .20±0, .04 P = 1-P M-P n, as explained i n section 2.E. -94-TABLE 5.2 Physical properties a f f e c t i n g electron escape from spurs, and P M, P D and P. values obtained f o r four pure l i q u i d s at 295 K. Liquid G . / G + a g i t b y c e vd 0 Ref. Neopentane 0.97 70 1.8 -0.39 0.18 0.55 0.27 This work c-hexane 0.99 0.4 1.8 +0.01 0.20 0.69 0.11 [33] Methanol 0.62 0.0006 33 NA 0.23 0.62 0.15 [42] Water 0.38 0.002 78 NA 0.20 0.62 0.18 [33,42] a Fraction of t o t a l radiation-produced electrons (Gt=4.5) which do not escape intraspur n e u t r a l i z a t i o n tq. become free ions. b M o b i l i t y i n cm 2V" 1s" 1 [103]. S t a t i c d i e l e c t r i c constant. ^ Conduction band energies of electrons i n the l i q u i d at 295 K. -95-ranging from c-hexane to water, with completely d i f f e r e n t p o l a r i t y , solvating power, and spur electron s u r v i v a l p r o b a b i l i t i e s , the muon y i e l d s are e s s e n t i a l l y the same. This leads one again [33] to conclude that the spur -model of muonium formation does not account f o r the muon y i e l d s i n d i f f e r e n t l i q u i d s . Instead, one i n c l i n e s to the view that Mu formation might depend on the p r o b a b i l i t y of hot Mu* emerging from the epithermal stages of the track without having undergone a hot atom abstraction or s u b s t i t u t i o n r e a c t i o n . Perhaps the p r o b a b i l i t y of such hot atom reactions occurring r e l a t i v e to thermalization i s very s i m i l a r f o r a l l saturated molecules composed of f u l l y hydrogenated C and 0 atoms. A second major i n t e r e s t l i e s i n the f a c t that the y i e l d s barely change at the l i q u i d - s o l i d phase t r a n s i t i o n , or with temperature over the range studied. For c e r t a i n , there i s no marked break i n the y i e l d s at the melting point. As shown by Table 5.3, t h i s i s i n sharp contrast to the r e s u l t s i n water [43] and the noble gases Ar, Kr, and Xe [86] (only the Ar data i s shown f o r comparison). Unlike neopentane, these other media show a complete absence of i n the s o l i d phase, with P^ increasing noticeably as the material was s o l i d i f i e d . P e r c i v a l [44] proposed that P^ i n water a r i s e s from the d e p o l a r i -zation of Mu by some paramagnetic species i n the muon's terminal spur. However, t h i s cannot explain the r e s u l t s i n Tables 5.2 and 5.3. An "expanding track model" was proposed by Walker [ 5 l ] to account f o r P^ i n non-polar as well as polar l i q u i d s . In t h i s model, thermalized Mu i s v i s u a l i z e d to stop some 2 ° distance (perhaps <_ 10 A) beyond the l a s t spur of the muon's track of i o n i z a t i o n and e x c i t a t i o n . However, as these paramagnetic and transient species d i f f u s e -96-TABLE 5.3 The e f f e c t of phase (and temperature) on the various muon y i e l d s i n neopentane, water and argon. Medium Phase T/K ?_L_ Ref. neopentane l i q u i d 295 0 .18 0 .55 0. .27 This work neopentane s o l i d 209 0 .19 0, .61 0, .20 This work water l i q u i d 295 0 .20 0 .62 0, .18 [43] water s o l i d 272 0 .52 0 .48 0 [43] argon l i q u i d 85 0 .48 0 .02 0, .50 [86] argon s o l i d 77 0 .91 0 .008 0, .08 [86] - 9 7 -apart, t h e i r expanding track overlaps with Mu, which i s d i f f u s i n g randomly. Within = 10 nsec, Mu then has a chance to react with these r a d i o l y s i s products i n the expanding muon track and r e s u l t s i n a loss of Mu-amplitude. Since the majority of these species are hydrated electrons (e ) and OH r a d i c a l s (OH') i n water, or free r a d i c a l s i n organic media, the track could have 2 ° -8 expanded to -10 A i n 10 sec at which time t h e i r concentration w i l l be -3 -5 2 -1 -10 M (calculated on d i f f u s i o n c o e f f i c i e n t s of 5x10 cm s ). Since spin conversions have k ^ - l O ^ M "^ s * [30], t h i s implies that there w i l l be a reasonably large p r o b a b i l i t y of reaction between Mu and these expanding _ 7 track species within <10 sec. This explains i n l i q u i d s such as argon (Table 5.3) at 85K, since d e p o l a r i z a t i o n of Mu by these track species w i l l cause a lowering of P.. and contribute to P.. In s o l i d s , the d i f f u s i o n should M L be s u f f i c i e n t l y slow that the expanding track requires >10 ^ sec to overlap with Mu, whose motion could also be suppressed at low temperatures or even be trapped i n some s i t e s of the s o l i d matrix. So, within the MSR time scale, one can observe t h i s P^ as P^ and therefore no missing f r a c t i o n appears. In view of t h i s expanding track model, one can now attempt to r a t i o n a l i z e P^ observed i n neopentane by assuming that the muon track s t i l l expands r a p i d l y in the s o l i d , and there may not be any trapping s i t e s a v a i l a b l e f o r Mu. For neopentane, t h i s i s reasonable since i t s s o l i d state i s of a p l a s t i c nature. According to Livingstone et a l . [104], neopentane behaves as a " l i q u i d - l i k e " c r y s t a l between -133 and -20°C. Indeed, the t r a n s l a t i o n a l and v i b r a t i o n a l motions of the l i q u i d state are s t i l l retained to a large extent i n the s o l i d state form even at these low temperatures. For instance, l i g h t p a r t i c l e s -98-such as Mu are expected to be j u s t as mobile i n s o l i d neopentane as i n the l i q u i d phase since i t was found that the electron m o b i l i t y i n p l a s t i c c r y s t a l s of neopentane was approximately twice as large as that i n the l i q u i d [105]. "This can explain the presence of i n s o l i d neopentane and i t s absence i n s o l i d argon and i c e based on the expanding track model. A t h i r d point of i n t e r e s t i n the above r e s u l t s i s the value of A Q obtained i n neopentane (see Table 5.1). This indicates that neopentane has a A Q - v a l u e close to the smallest found f o r any system at TRIUMF or SIN (see chapter 3). Indeed, i t i s the same as that found f o r water i n chapter 3. As i n section 3.A.2., considering a A . . . contribution of 0.09x10^ s \ A f o r ' 6 AH o neopentane becomes A < 0.17 x 10 6 s " 1 . (5.1) o — 4 -1 -1 Using t h i s value of A , one obtains k„ < 2x10 M s f o r the abstraction b o M reaction, Mu + C ( C H 3 ) 4 > MuH + C(CH 3) 3CH 2 . (5.2) By analogy, the corresponding H-atom reaction has a rate constant, k^, of 1x10^ M~*s"* [106]. The smaller k^ value f o r Mu as compared to H was explained by the p o s s i b l e endothermicity of reaction (5.2) [107]. In contrast the corresponding H-atom i s exothermic (~1 kJ mole" 1) [108]. This d i f f e r e n c e stems from the higher zero-point energy of Mu-bonds compared to H-bonds. However, i f -99-one considers the Mu-FB as i n chapter 3, one can probably conclude, as i n the case of X q i n water, that Mu does not react i n neopentane and that the X Q-value obtained i s simply an experimental a r t i f a c t due to physical phenomena i n the aforementioned chapter. Similar magnetic f i e l d experiments to investigate X q i n neopentane are being planned f o r future beam times at TRIUMF. 5.B. Muon y i e l d s i n concentrated OH solu t i o n s . It was noted by Pe r c i v a l et a l . [43] that the reaction, Mu + OH" ?• e" + MuOH (5.3) aq aq 7 -1 -1 has a rate constant at room temperature of 1.7x10 M s (the k i n e t i c s of t h i s r e a c t i o n were investigated and w i l l be discussed i n chapter 6). This rate means that Mu i n hydroxide solutions having pH greater than 12 w i l l not be observable on the MSR timescale of about 10 ^  sec. Also, as no Mu-radicals are produced i n KOH/H^O systems, the only y i e l d at high pH i s the diamagnetic f r a c t i o n , P^. This section w i l l give r e s u l t s of muon y i e l d s at high OH concentrations which have bearing on the v a l i d i t y of both spur and hot models of muonium. 5.B.I. Results. Experiments with the highly concentrated OH solutions were done at 80G at room temperature. Analar grade KOH p e l l e t s were dissolved i n t r i p l y d i s t i l l e d deionized water. The concentrations of solutions were checked a f t e r -100-d i l u t i o n by t i t r a t i o n with a standard acid, potassium hydrogen phthalate. The mole, mass, and volume f r a c t i o n s were evaluated from the measured masses of KOH and ^ 0 used to make up the so l u t i o n s . For these experiments, the -solution was contained within two sheets of polyethylene (3 mil) i n an other-wise regular t e f l o n c e l l set-up (Figure 2.11a). The need f o r polyethylene was that 2-20 M KOH solutions react c o r r o s i v e l y with mylar sheets. Surface muons were used f o r the i n v e s t i g a t i o n and the asymmetries were normalized (equation 2.11) against that of CCl^ i n the same sample holder. In the case of pure KOH, the c e l l was f i l l e d with KOH p e l l e t s , where only a minute f r a c t i o n of the muons would come to re s t i n the walls of the vessel or i n the gas phase. The presence of some carbonate i n the surface layer of the p e l l e t s exposed to a i r could be a source of uncertainty i n that measurement. The muon y i e l d s (Pp) were found to change with the mole/mass/volume f r a c t i o n s of KOH i n the manner shown i n Figure 5.2. Pp i s seen to increase from 0.62 to 0.79 as the medium changed from pure water to ca_ 50% KOH. S o l i d KOH p e l l e t s gave a P^ reading of 0.55. 5.B.2. Discussion. The A D values that one observes i n a ySR experiment can only a r i s e from contributions to one's histograms i f they correspond to muon frequencies (to^ ) precessing with coherent i n i t i a l phases. For example, equation (2.15) describes the amplitude contribution to A(t) from a l l muons which are ( i ) i n i t i a l l y placed i n stable diamagnetic species (y + , MuOH and MuH) during aq thermalization i n the muon track and ( i i ) also those diamagnetic species formed -101-0.0 0.2 0.4 0.6 0.8 1.0 Fraction of KOH in water Figure 5.2: V a r i a t i o n of Pp with composition of KOH/f^O so l u t i o n s , p l o t t e d as mole ( A ) , volume ( • ) , and mass (•) f r a c t i o n s . The dashed l i n e i s the s t r a i g h t l i n e connecting Pp i n pure l i q u i d H,0 with pure KOH p e l l e t s . -102-a f t e r thermalization - but only by reactions which are s u f f i c i e n t l y f a s t to avoid dephasing by precession at another frequency (or depolarization due to magnetic i n t e r a c t i o n s ) . Thus, f o r muons which i n i t i a l l y appear as Mu atoms, "they must be converted to diamagnetic species (such as MuOH) v i a chemical reaction with OH before dephasing or depolarization sets i n , i f they are to contribute to at a l l . In the case of Mu atoms with p a r a l l e l electron and T muon spins (the s o - c a l l e d " t r i p l e t s " , Mu) coherence o f the muon spin w i l l be l a r g e l y retained i n the product i f the reaction rate greatly exceeds the differen c e between the Mu and muon frequencies - which changes with magnetic f i e l d . For the " s i n g l e t s " ( Mu), however, chemical conversion must precede depo l a r i z a t i o n of the muon spin by o s c i l l a t i o n s at the hyperfine frequency, COq [109]. This i s independent of magnetic f i e l d below the "Paschen-Back" region (the region between co (~cog)> COq i n Figure 2.5). In addition, Mu atoms can be " l o s t " (P^) due to depolarization by encounters with r a d i a t i o n produced species during the non-homogeneous expansions of the terminal spur or of the e n t i r e track. S For reaction (5.3), since k^ i s r e l a t i v e l y small, a l l Mu w i l l be depolarized by the hyperfine frequency (2.8x10"^ s *) because i t s t i l l - - 8 -1 greatly exceeds the value of k M[0H ] even at 20 M ( i . e . k^[0H ] = 3.5x10 s ). T 8 - 1 However, at 80G, Mu depolarization (co^ = 6.9x10 s ) covers the very time-scale over which reaction (5.3) proceeds. In f a c t , one can already predict q u a l i t a t i v e l y that P Q should increase i n these concentrated KOH systems from "h" to a maximum of about [h+0.5(1-h)]: i . e . from 0.62 to 0.81, i f h remains constant at 0.62. In t h i s t h e s i s , h represents the f r a c t i o n of muons which -103-form stable diamagnetic species (before thermalization) v i a hot-atom abstraction or s u b s t i t u t i o n reactions during c o l l i s i o n a l deexcitation following charge-exchange, or by thermalization as y + ions. One must d i s t i n g u i s h t h i s use of h from h^, which r e f e r s to a much l a t t e r point i n time and was introduced to describe the y i e l d of diamagnetic f r a c t i o n s emerging from the terminal spur i n the spur model of Mu formation [43,52], With regard to Mu formation, one has the following expectation. As the proton r e a c t i o n [5.4], H + + OH" * H 20 (5.4) i s one of the f a s t e s t reactions known, with a rate constant of 1.4x10"''"'' M * [110], one would expect the analogous reaction of y + with OH (5.5) to be f a r too f a s t f o r y + to f i n d an e i n a y + + OH" > MuOH (5.5) spur f o r Mu formation by thermal reacti o n . Therefore, one would expect that P D f o r 2, 5, 10 or 20 M KOH a l l be 1.0 on the basis of the spur model. S i m i l a r l y , i t was noted previously be Walker et a l . [48] that i f the spur model should apply, then the observable Mu y i e l d should have been reduced equally by y + scavengers as by e scavengers, but no such reduction was found. As one can see from Figure 5.2, P^ does not approach 1.0 even at ~20M 0H~. This implies that the spur model does not apply f o r these systems. -104-We now continue the discussion of these r e s u l t s i n terms of a hot model, one which describes h as the i n i t i a l (hot) y i e l d of diamagnetics and (1-h) as the i n i t i a l y i e l d of Mu. Various formalisms f o r the q u a l i t a t i v e treatment of the exact time-dependence of muon p o l a r i z a t i o n s have been given i n r e f e r -ences [4, 10, 16-18, 28]. Following Fleming et a l . [28], the t o t a l muon p o l a r i z a t i o n (Pres) extrapolated back to t=0 (the time at which Mu i s formed) i s given by equation (5.6), Pres = lim P(t) exp(-ia> Dt) (5.6) t-x» where P(t) represents the time dependence of the en t i r e muon ensemble including T S Mu and Mu converted to D. This P ( t ) , of course, depends on the model of Mu reactions. In t h i s chapter, the following model i s proposed: u o » Depol.(hyperfine) track M * [0.5(1-h)] T M > Dephasing (5.7) V A A A M ; >Mu != - J-+-=> Mu _ depol. (track overlap) where Mu* i s a hot Mu atom and X=k^[0H ] f o r reaction (5.3). The dotted l i n e represents depolarization of Mu (P^) v i a track overlap [51] and w i l l not be included i n the following d e r i v a t i o n of P Q f o r reasons that w i l l be explained l a t e r . Equation (5.7) i s a single-step Mu reaction model (conver-T S sion o f Mu at 80G), since a l l Mu w i l l be depolarized through the hyperfine o s c i l l a t i o n s before reaction to D (as to »\). so that h a l f of the Mu ensemble o » -105-is disregarded. By applying the complex muon polarization in free Mu-atoms at low fields (B<150G) [75], one obtains the total observable asymmetry as, t ApCt) = ApCo) + A J J C O ) / A-exp - (A+iAiot') d t 1 (5.8) o where Ato=co D-co^--co M . Integration of this equation to t=» (since observation takes place long after the rapid motions of the y + spin in Mu have settled down to the slow precession of the free y +-spin in a diamagnetic environment) for the Mu ensemble, followed by normalization to fractional yields, gives equation (5.9). (See Appendix II). P D = h + 0.5.(l-h) {A/(A2 + co M 2) - i i} • (5.9) Probably one would not expect h and k^ to be the same throughout the 0-20 M concentration range, but equation (5.9) is shown by the dashed line in Figure 5.3, calculated by keeping h=0.62 and k^=1.7xl07 M *s 1 for a l l concentrations. It can be seen very clearly that P calculated by equation (5.9) with h and k^ fixed does not correspond to the experimental values. T There is a further channel by which Mu may be lost (dashed line in equation T (5.7)) represented by depolarization through intra-track encounteres of Mu with paramagnetic species, as mentioned earlier. Such encounters are most -8 important ~10 sec after Mu formation, as the expanding track overlaps with Mu formed some 10 nm beyond i t [51]. This extra decay mode w i l l reduce Pp, but i t is lik e l y to compete with the conversion of Mn+D mainly in the lower portion of the concentration range. Therefore, this mode of Mu log([KOH]/M) Figure 5.3: Plot of Pp against log [KOH]. The dashed l i n e i s calculated using equation (5.9) with h and k^ constant (see t e x t ) . The s o l i d l i n e i s that used to ' f i t * to equation (5.9) with k^ or h variable - thereby giving the values i n Table 5.4. -107-depolarization cannot account for the observed being significantly larger than that calculated through equation (5.9) at the high concentrations. However, i t is interesting to note that i f one allows the value to -change with [OH ] at greater than 1 M concentration, one finds that the data readily f i t those of equation (5.9). A change of k^ at very high concentra-tions i s expected in kinetic studies [ i l l ] . As shown by Table 5.4, in order for agreement between experimental and calculated P^ , k^ has to increase to 7 - 1 - 1 about 7.0x10 M s at 20 M KOH. This four-fold change in is certainly not unreasonable for a solution in which the mass fraction of KOH reaches the 0.6 level. Alternatively, one could have obtained a good f i t of the data to equation (5.9) i f one allows h to change. The required variation of h with concentration is given also in Table 5.4. One would certainly expect h to change since the structural and chemical properties of water vary quite drastically when one third of i t s molecules are replaced by KOH. Therefore, by recognizing these factors at high concentrations, i t is reasonable to conclude that the hot model given in equation (5.7) is not inconsistent with the observed enchancement in P^. 5.C. Conclusion. The muon yields in neopentane and concentrated aqueous KOH solutions point toward a general conclusion; that i s , the findings seem to be at variance with the expections of a spur model of Mu formation. -108-TABLE 5.4 Variation of k M or h required for equation (5.9) to f i t the experimental P~ data. [0H"]/M Pp (±0.01) k M a/10 7M" 1s" 1 h' b 0 0.62 1.7 0.62 2 0.62 1.7 0.61 5 0.64 1.7 0.62 10 0.70 3.3 0.66 20 0.79 7.0 0.73 Footnotes: k^ are the values of k^ obtained by ' f i t t i n g ' the experimental data to equation (5.9), by keeping h constant at 0.62 but allowing k M to change with [OH ] b h' are the values of h (hot fractions) necessary to get the 'fitted' 7 -1 -1 solid line i f k w is invariant at 1.7 x 10 M s -109-CHAPTER 6 MUONIUM KINETICS IN AQUEOUS SOLUTIONS -110-In the M.Sc. t h e s i s , the temperature dependence of various s o l u t e s r e a c t i n g w i t h Mu was i n v e s t i g a t e d . I t has been shown th a t Mu can undergo many types of r e a c t i o n s , such as a b s t r a c t i o n , r e d u c t i o n , acid-base t r a n s f e r , a d d i t i o n , s u b s t i t u t i o n , and spin-conversion. With the establishment of a muon community at TRIUMF, many of these r e a c t i o n s i n v o l v i n g water s o l u b l e i n o r g a n i c and organic compounds were stud i e d [ 3 ] , As mentioned i n chapter 1, the comparison between Mu to H atom r a t e constants i s an important step i n understanding of k i n e t i c isotope e f f e c t s , quantum mechanical t u n n e l l i n g and H atom r e a c t i o n s i n g e n e r a l . In t h i s chapter, two r e a c t i o n s w i l l be discussed. S e c t i o n A describes the acid-base r e a c t i o n of Mu with hydroxide ion (OH^), while s e c t i o n B e n t a i l s the Mu a b s t r a c t i o n r e a c t i o n with deuteroformate ions (DC0 2~). Both of these w i l l be compared to the HC0 2~ r e s u l t from the M.Sc. work. 6.A. Muonium r e a c t i o n with OH a q In a l k a l i n e s o l u t i o n s , H atoms are converted i n t o a species which r e a c t s with scavengers i n the same manner as r a d i a t i o n produced e [112]. The aq r e a c t i o n , H + OH" 9 e + H o0 (6.1) aq aq 2 was i n v e s t i g a t e d by various groups [113, 114] i n which i t was i d e n t i f i e d as a proton t r a n s f e r r e a c t i o n . A l s o , i t was used to evaluate the pK a of the hydrogen atom [115] and i t has some general i n t e r e s t s i n c e i t demonstrated that -111-H and e constituted a conjugate acid-base pair [116]. The rate constant aq (k^) for equation (6.1) was determined by Neta et a l . [117] as having a 7 -1 -1 value of 1.5x10 M s . When compared to the analogous reaction for Mu, Mu + OH" > e" + MuOH ; (6.2) aq aq 5 there was no kinetic isotope effect found [43, 87, 118]. This exact correspon-dence in rates therefore created an interest to study the temperature dependence of this "muon-transfer" reaction. 6.A.I. Results. MSR measurements were carried out for four separate temperatures ranging from 1 to 84°C. Surface muons and the temperature c e l l (Figure 2.11b) were employed to extract k^ according to equation (2.16) using various values of [OH ], with the actual A q obtained in a separate experiment on pure water at that temperature (as A q changes slightly with temperature [71]) . It should be noted that in Mu solution kinetics,, since MSR measurements are accumulated over 4 muon lifetimes, solute concentrations must be chosen to bring A 1 into the most accurate part of that range, namely -lysec. Figure 6.1 shows MSR histograms for A Q and A ^ at 59°C for pure water and 0.01 M KOH, respectively. Table 6.1 gives the k^'s obtained for each of the four temperatures using different OH concentrations. These results are plotted (Figure 6.2) using the familiar Arrhenius expression (equation (2.17)) to obtain values of E and A, which are given in Table 6.2. -112-42000.0 30000.0 h c o 18000.0 u 6000 .0 0.30 E E in < 0.20 h 0.10 h o . o o h -0 . 10 1 2 3 T ime in jxsec (S n s e c / b m ) Figure 6.1: Typical MSR histograms showing the muonium signal at 8 gauss f or (a) pure water at room temperature and (b) i t s decay at 0.01 M KOH at 59 C. The l i n e drawn i s the computer's best f i t of the data points to equation (2.15). -113-TABLE 6.1 Second order rate constants (k M) obtained from using equation (2.16) at various temperatures and OH concentrations. [OH"]/10"3M Temp/°C A M/10 6s" k kM/M"1s"1 1.60 84 0.97 3.8 x 10 8 7.49 59 1.62 1.7 x 10 8 49.9 22 1.15 1.7 x 10 7 82.5 1 0.93 7.8 x 10 6 -114-2 1 . 0 20.0 h 19.0 h 18.0 h 17.0 h 16.0 h 15.0 h 14.0 2.50 3.00 3.50 103 °K/T 4.00 Figure 6.2: Arrhenius p l o t showing InCkjP versus T 1 for the Mu+OH reac t i o n . -115-TABLE 6.2 Arrhenius parameters obtained f o r k M at ~ 295 K, and comparison with (i) k„ [99], and ( i i ) reaction of Mu with HCO " [70]. Reactant k M/M" 1s" 1 E a / k J mole" 1 A/M _ 1s" 1 kM / kH OH' 1.7 x 10 7 40 ± 5 (2.4±0.1)xl0 1 4 1.1 ±0.2 HC0 2" 3.4 x l O 6 33 ± 2 4 x 1 0 1 2 0.01 -116-6.A.2. Calculations and Discussion. That there i s no k i n e t i c isotope e f f e c t (k^/k^) found f o r the OH reaction at room temperature (see Table 6.2), the only temperature where there i s H data, agrees with the previous f i n d i n g [43]. It suggests that the reaction mechanism i s of the same type f o r Mu as f o r i t s H analogue (compare equation (6.1) and (6.2)), i n which the net e f f e c t f o r Mu i s the transference of a muon to the powerful base: muonium behaving as a "muonic a c i d " . 7 -1 -1 The k^ value of 1.7x10 M s at room temperature f o r the reaction of Mu with OH indicates that i t i s not a d i f f u s i o n c o n t r o l l e d reaction, since ^ d i f f v a l u e s °f ~ 2 x l 0 ^ M *s * are u s u a l l y found [3,99]. Certainly, the observed a c t i v a t i o n energy of 40 kJ mole * (Table 6.2) r e f l e c t s that the r e a c t i o n greatly exceeds the E f o r d i f f u s i o n of species i n water, which i s generally about 17 kJ mole * [54,70]. Further, f o r d i f f u s i o n c o n t r o l l e d reactions, a p l o t of k^/T against the inverse of v i s c o s i t y should y i e l d a s t r a i g h t l i n e (see Smoluchowski's equation (1.10) i n section l . C . l . ) . This a r i s e s since k^ should be proportional to the r e l a t i v e d i f f u s i o n constants, which i n turn would show the temperature dependence of the v i s c o s i t y [123,120]. Table 6.3 gives the v i s c o s i t y (n) as a function of k^/T, and as one can see from Figure 6.3, the p l o t of k^/T versus 1/n does not y i e l d a s t r a i g h t l i n e . This corroborates the f a c t that the Mu reaction with OH i s not d i f f u s i o n aq c o n t r o l l e d . Instead, i t i s presumably a c t i v a t i o n c o n t r o l l e d , as evident by the large a c t i v a t i o n energy obtained. Following equation (1.1) i n section l . C . l . , i t i s convenient to construct the following reaction scheme: -117-T A B L E 6 . 3 V i s c o s i t y p a r a m e t e r s a s a f u n c t i o n o f t e m p e r a t u r e s f o r t h e r e a c t i o n o f Mu w i t h O H " . a q T / ° C (k M/T)/M" 1s" 1K" 1 ( l / n ) + / 1 0 - 2 p o i s e s 84 1 . 0 5 x 1 0 6 2 . 9 6 59 5 . 1 2 x 1 0 5 2 . 1 1 22 5 . 7 6 x 1 0 4 1 . 0 5 1 2 . 8 4 x 1 0 4 0 . 5 8 n i s t h e v i s c o s i t y , o b t a i n e d f r o m r e f e r e n c e [ 1 1 9 ] . -118-12.50 10.00 CO o 7.50 r 5.00 -2.50 0.00 0.0 1.0 2.0 3.0 (l/77)/lO~2poises Figure 6.3: Plot of k^/T versus the inverse of the v i s c o s i t y f o r the Mu+0H" reaction . -119-k d Mu + OH" x [Mu . . . OH ] 2 => MuOH + e (6.3) to describe bimolecular reactions i n s o l u t i o n . The rate constants (k,, k § d p k^) are as described previously (see section l . C . l . ) . In t h i s case, the encounter-pair i s the [Mu...0H ] complex formed by the d i f f u s i o n of the two reacting species and caged momentarily by the solvent. Applying the steady state approximation to t h i s encounter-pair, one obtains, as i n equation (1.5), k^ equalling the composite of three parameters, k^k^/(k^+k^). Since the OH reaction i s << k ^ ^ , k^>>k^ (encounters seldom leading to re a c t i o n ) , one obtains k^ - k^k^/k^. Thus, the observed Arrhenius parameters may also be expected to be composites of: A = A^A^/A^ and E g = E^E^-E^. I f one considers the establishment of the encounter-pair as equivalent to a pseudo-equilibrium c o n t r o l l e d mainly by d i f f u s i v e processes, then E^ and E^ should be small, and i n any event comparable to each other i n magnitude. In e f f e c t , E approximates E^, the a c t i v a t i o n b a r r i e r of the slow r a t e - c o n t r o l l i n g step. With these considerations pertaining to reaction (6.2), one sees that the muon-transfer step i s characterized by an E & of 40 kJ mole However, the unusually large value of A found (see Table 6.2), being a composite of ApA^/A^, perhaps r e f l e c t s the unique as s o c i a t i o n of OH with the water structure. It i s unfortunate that the Arrhenius parameters f o r the analogous H atom reaction (6.1) do not seem to be av a i l a b l e f o r comparison. Instead, i t i s o f i n t e r e s t to compare t h i s Mu+0H~ reaction with that of the Mu+HC02 reaction as given i n Table 6.2. Here both E & and A are smaller than with OH and there i s a large (inverse) isotope e f f e c t . Towards formate, Mu i s expected to ^ -120-abstract the H atom - a reaction type i n which H i s generally much more e f f i c i e n t [3]. Indeed, one would not expect formate to be a s u f f i c i e n t l y strong base to force Mu to undergo an acid-base t r a n s f e r as i n reaction (6.2). 6.B. Muonium abstraction with DCO,, . In the M.Sc. t h e s i s , the HCC^ r e a c t i o n was studied as a function of temperature. For comparison, the DCO^ abstraction reaction w i l l be described i n t h i s s e c t i o n . With these r e s u l t s , not only can one compare the isotope e f f e c t i n which the isotope i s the attacking species, but also the primary isotope e f f e c t i n which the isotope i s the atom to be transferred (abstracted). Abstraction reactions by hydrogen atoms have been well studied over the years. Indeed, Anbar et a l . [121] have investigated H-D isotope e f f e c t s on hydrogen abstraction by H atoms from 2-propanol, formate, methanol, and acetate. They concluded that the H-D isotope e f f e c t may be c o r r e l a t e d with the r e a c t i v i t y of the given compound toward hydrogen abstraction by H atoms and speculated about the contribution of a t u n n e l l i n g e f f e c t f o r these reactions. Now with Mu, t u n n e l l i n g could be even more manifest due to the smaller mass. Therefore, a temperature-dependence study of Mu+DCC^ can contribute towards furt h e r understanding of isotope e f f e c t s i n s o l u t i o n k i n e t i c s . 6.B.I. Results. Surface muons were again used together with the temperature c e l l f o r these measurements. A l l DCC^ solutions were obtained by t i t r a t i o n of formic-acid-d., (98 atom% D) with NaOH of known concentration, u n t i l the pH was about 7. - 1 2 1 -Table 6.4 and Figure 6.4 d i s p l a y the values of A M obtained f o r d i f f e r e n t concentrations of DCC^ at 80°C. The slope of the l i n e gives i n accordance with equation (2.16). Values of k^ obtained i n t h i s way f o r the d i f f e r e n t temperatures are given i n Table 6.5. These are then p l o t t e d i n accordance with the usual Arrhenius treatment (Figure 6.5) to y i e l d : E = (39.0±3.2) kJ mole 3. and A = (1.1± 0.2) x 1 0 1 3 M' 1s' 1. 6.B.2. Discussion. As with the OH reaction, the a c t i v a t i o n b a r r i e r f o r the DCO^ reaction i s f a i r l y large. By comparing i t to the analogous H atom reactions, the mechanism i s presumed to be the abstraction of deuterium from DC0 2 . Mu + DC0 2" > MuD + C0 2" . (6.4) With such a low k^ (-1x10^ M *) at room temperature, one expects t h i s reaction to be a c t i v a t i o n c o n t r o l l e d . Indeed, a p l o t of k^/T versus 1/n (see previous section) gives a non-linear curve s i g n i f y i n g that the reaction i s not d i f f u s i o n c o n t r o l l e d . Compared to the reaction of Mu abstracting H from HC0 2 [70] (see Table 6.6), reaction (6.4) i s expected to have a smaller k^ than HC0 2 due to i t s larger E (39 kJ m o l e - 1 ) . This can be q u a l i t a t i v e l y understood i n terms of the height of a v i b r a t i o n a l l y adiabatic t r a n s i t i o n - s t a t e - t h e o r y b a r r i e r [70]. Table 6.6 shows that as the r a t i o of masses o f abstracting/ abstracted atom decreases, zero-point energy changes i n the reactants and -122-TABLE 6.4 Muonium decay rate constants O m) as a function of DC02 concentration at 80°C. [DC0 2 ]/10" 3 M 0.0 ± 0.0 44.5 ± 1.0 62.2 ± 1.0 88.9 ± 1.0 0.36 ± 0.01 1.80 ± 0.64 1.30 ± 0.30 2.13 ± 0.52 -123-2.80 5 0 .80 (D 0.40 0.00 0 10 20 30 4 0 50 60 70 80 90 100 DC0 2 ~ C o n c e n t r a t i o n / 10~ J M Figure 6.4: Plot of X M versus DC02~ concentration at 80 C. -124-TABLE 6.5 Second order rate constants (k M) obtained from A^ using equation (2.16) at four temperatures f o r the Mu + DCC^ r e a c t i o n . Temp/°C k /M V 1 _M 1 (4.3 ± 0.8) x 10 5 22 (9.9 ± 2.3) x 10 5 60 (8.5 ± 0.8) x 10 6 80 (1.8 ± 0.3) x 10 7 -125--126-TABLE 6.6 Comparative k i n e t i c parameters f o r various abstraction reactions, Reaction #k/M" 1s" 1 m1/m.+ k,./k k'/k' E /kJ mole 1 A/M" 1s" 1 1 2 M H H D a 5.2 H + HC0 2" 1.2 x 10 8 1.00 H + DC0 2" 2.3 x 10 7 0.50 Mu + HCO " 3.4 x 10 6 0.11 0.03 % 33.4 ± 2.3 (4.1±0.2)xlO 1 2 5 3 - 4 13 Mu + DC0 2 9.9 x 10 0.056 0.04 > 39.0 ± 3.2 (l.l±0.1)xl0 m^  i s the mass of the attacking atom (Mu or H) and m2 i s the mass of the abstracted atom (H or D). # k are the room temperature values, the H-atom values are from reference [99]. --127-t r a n s i t i o n - s t a t e s e f f e c t i v e l y cause a higher b a r r i e r , i n l i n e with the observed trend i n reaction rates. However, i t should be noted that (as pointed out i n the M.Sc. thesis and reference [70]) the order of reaction rate i s also the order of decreasing endothermicity. The trend i n rates indicates that the p r o b a b i l i t y of a l i g h t atom abstracting a heavier one i s smaller than that of an atom abstracting an equivalent sized atom. The trend follows ones expectations based on simple mechanics of c o l l i s i o n s . Table 6.6 demonstrates two separate isotope e f f e c t s , ^ / k ^ r e f e r s to the case when the atom being abstracted i s d i f f e r e n t , while k^/k^ i s f o r the case when the attacking atom i s d i f f e r e n t . The trend f o r these two isotope e f f e c t s are i n opposite d i r e c t i o n s (e.g. k^/k^<l, and k^/k^ > 1) . In the case of k,,/k„, Roduner and Fischer [122] have concluded that t h i s reverse M H isotope e f f e c t i s due to quantum mechanical t u n n e l l i n g i n the abstraction r e a c t i o n . However, i t was argued [70] that such a t u n n e l l i n g e f f e c t should also decrease the A-factor (as i n NO^- [70]) and lower the observed a c t i v a t i o n energy. In the case of k'/k', smallish isotope e f f e c t s (5.2 § 3.4) were n 1J obtained f o r the H and Mu abstractions from HC0 2~ and DCC>2~. As mentioned i n section l.C.3., quantum mechanical t u n n e l l i n g should give r i s e to curvature in Arrhenius p l o t s , apparent decrease of A-factors (due to small E 's) , and large k i n e t i c isotope e f f e c t s . That none of these e f f e c t s are apparent f o r the HC0 2~ and DC02~ implies t u n n e l l i n g does not dominate the abstraction of H and D atoms i n HC02~ and DC02~ by e i t h e r H or Mu. It i s unfortunate that water has such a short l i q u i d range i n which to study isotope e f f e c t s with solutes. This l i m i t a t i o n could prevent the detection - 1 2 8 -of manifestations of quantum mechanical t u n n e l l i n g . Perhaps t h i s r e f l e c t s the need to use non-polar media, such as hydrocarbons, some of which have much larger l i q u i d ranges, to investigate k i n e t i c isotope e f f e c t s of Mu with various solutes. -129-CHAPTER 7 COLLABORATIVE WORKS -130-A l l projects up to and including chapter 6 are experiments of my own undertaking. In the present chapter, however, various other projects w i l l be b r i e f l y reviewed which are investigations done in collaboration with -Dr. Y. Ito § Mr. Y. Miyake (hydrocarbons), Dr. Y.C. Jean (model biological systems § solvent mixtures), and Dr. J.M. Stadlbauer (Mu kinetics of solutes in aqueous solutions). A l l of this work has now been published in various journals [33,34,36,38,39,40,41,49,74,93,123]. These projects were colla-borative in the sense that the aforementioned research-associates were major investigators. Due to the nature of the TRIUMF beam-time system -one in which each group of experimenters is given on average one week of experimental time every 4 months - a l l projects must be done in close collaboration, with many people involved. The brief description of these projects reflects my relatively minor role in them. 7.A. MSR applications to model biological systems. As mentioned in chapter 1, the MSR technique is being applied to study the chemical interactions of Mu atoms with some model biologically-important systems. For the M.Sc. thesis [7l], the reaction of Mu with porphyrins (hemin and protoporphyrin [37]) was found to consist of a peripheral site attack at the Tr-system and of the central ion. The mechanisms were mainly through addition to the conjugated double bonds for the protoporphyrin and by reduction or partial spin conversion processes for the hemin solutions. For this Ph.D. thesis, other systems are being reported, namely micelles [39] and cyclodextrins [38]. An advantage of this MSR technique over some other -131-conventional techniques in biological studies is that i t is a non-destructive method. Furthermore, since H atoms are involved in many fundamental biolo-gical processes, such as charge transfer, proton transfer, hydrogen bonding, etc; i t follows that Mu atoms are ideal probes with which to study these complicated systems. Consequently, Mu can act as a direct probe inside giant molecules and extract information about the site of electron transfer and spin state of metallic ions. The f i r s t indication of this potential with Mu as a probe in biological application was revealed when Mu was used as a test of the spin state of transition metal ions in simple inorganic complexes in solution [30]. Perhaps the results given in this chapter w i l l stimulate the biochemists and biophysicists interest in u t i l i z i n g MSR as a means to study chemical dynamics in biological systems. 7.A.I. Muonium reactivity in cyclodextrins. Cyclodextrins are doughnut-shaped molecules, formed by linking several units of D(+) glucose together in a ring. In this experiment, a and 0-cyclodextrins were used. These are composed of six and seven glucose units, o respectively. Each has a molecular diameter of 8 to 9 A with central o cavity radii of 5 and 7 A. In aqueous solutions, they can support two distinct environments - the internal hydrophobic face along the central void and the external hydrophilic face on the sheath surface and beyond. They are well known for their a b i l i t y to host various kinds of guest mole-cules, creating inclusion compounds that are important for catalysis or inhibition in many chemical and biological reactions, and also as models of enzyme active sites [l24]. -132-Surface muons were used for the MSR experiments. Chemicals were bought from the Aldrich Chemical Co. (highest purity) and were used without further purification. A l l samples were prepared fresh before experiment; in parti-cular, iodine crystals were sublimed from analytical-grade reagents. The Mu rate constants (k„) with I„ and I, (I-+KI) in various solutions are M 2 3 2 shown in Table 7.1. By comparing to the analogous H atom reaction [99], the reactions in a l l media are presumed to be, Mu + I 2 > Mul + I (or u + + I 2") (7.1a) and Mu + I ' > y + + i " + I ~ . (7.1b) From Table 7.1, one can see that there is a tremendous increase of k„ M (Mu + I 2) as the medium is made more hydrophobic (i.e. in the sequence water methanol -> n-hexane) . However, when the I 2 is inside the hydrophobic enclosure of the cyclodextrin the value of k^ is almost unchanged from that of water. In fact, i t is slightly reduced although not much beyond the given error bars. For the I^ reaction, there is a two- or three-fold decrease in kM« Notice that Mu does not react significantly with either cyclodextrins or I . The fact that Mu reacts faster with I^ than with I 2 may be due to the larger size of I^ , particularly when one considers i t s coordination shell. This is expected since both reactions have k^ s corresponding to diffusion controlled rates [43]. The decrease of k M when these are incorporated as -133-TABLE 7.1 Muonium reaction rate constants (k M) with I 2 and I 3 ~ in various solutions at 295 K. Solution ( [ I 2 ] = 0.088 mM) . / i n10..-l -1 k„/10 M s M in water 1.7 ± 0.3 l2 in methanol 16 ± 6 h in n-hexane 29 ± 10 + a-cyclodextrin (1 mM) in water 1.2 ± 0.3 h + B-cyclodextrin (2 mM) in water 1.4 ± 0.3 h + I~ (0.16 mM) in water 5.9 ± 1.2 l2 + I~ (0.2 mM) + a-cyclodextrin (1 mM) in water 2.5 ± 0.5 l2 + I" (0.2 mM) + g-cyclodextrin (2 mM) in water 2.1 ± 0.5 a- or 3-cyclodextrin in water -0.002 I" in water 0.007 ± 0.001 -134-guest molecules in cyclodextrins seems to reflect the shielding effect of the solute from encounters with Mu. It may also imply that there are signi-ficant van der Waals forces present between the guest (l^ and I^ ) and host --molecules which causes the reduction in k^. In any case, the cyclodextrin encasement does not seem to transmit Mu as i f i t were an active membrane-like enclosure. This is evident from the fact that in the presence of a hydrocarbon medium reacts extremely fast with Mu relative to that of water, and since the inside of the cyclodextrins have a hydrophobic environment, one could have expected an increase in rate. Although i t is too early at this stage to extract formation constants from these data as in the case of posi-tronium studies [125], one can at least conclude from these results that reactions are curtailed to a certain extent in these macromolecular sugar systems. 7.A.2. Muonium reactivity in micelles. Micelles are formed by aggregation of surfactant molecules in aqueous solutions. These aqueous micellar systems contain two distinct environ-ments, an internal hydrocarbon-like core and an external water-like shell. The importance of these systems has been recognized in many biological and industrial applications [126]. They can be used as the simplest models with which to investigate mechanisms of bio-membranes and energy storage and conversion media. In these experiments, sodium octyl sulphate (NaOSA) was used above i t s c r i t i c a l micelle concentration (CMC) to provide a large (-50 monomer units) -135-internal hydrocarbon environment and negatively charged hydrophilic outer surface. Five solutes were chosen for the study, two of which ( N i 2 + , 3_ Fe(CN), ) w i l l be dissolved in the outer surface of the micelles and the o others f ^ , phenol, and naphthalene) are much more soluble in hydrocarbon media than in water, so they should have been present only within a micelle. The values of k^ were determined for these solutes in pure water, in water containing NaOSA below i t s CMC, and in water containing an excess of micelles (above CMC), k^ was measured for the reaction between Mu and 6 1 1 NaOSA when a value of 2x10 M s was found; this is a negligible rate and cannot significantly contribute to k^ in these experiments. In fact, A q for equation (2.16) was taken to be the observed X in the micelle solution in the absence of any solutes. The Mu reaction rate constants with these solutes in water, an organic solvent, and in NaOSA micellar aqueous solutions are given in Table 7.2. 2+ 3-For Ni and Fe(CN)^ , there was no observable change in k^ before and after CMC. This is expected since these two solutes are dissolved in the bulk aqueous medium, or the Guoy-Chapman region of the micelle. However, in the case of 1^ and phenol, a substantial increase (about two-fold) was found for k^ above CMC, and for naphthalene an increase of k^ by a factor of 6 was found above CMC. By comparing with data of H atom and positronium, the reactions of Mu with 1^ and phenol are at the diffusion controlled limit, whereas naphthalene is not. Perhaps the increase of k^ above CMC is due to a caging effect for these three solutes, and since k^ for naphtha-lene is in an activation controlled region, the observed increase can be greater for this solute than for I ? or phenol. -136-TABLE 7.2 Reaction rate constants k M/10 1 0 M _ 1s" 1 of muonium in water, an organic solvent, and in NaSOAa micellar aqueous solutions, at 295 K. (in CH3OH k^(above) Solute (in water) or hexane) (below CMC) (above CMC) k (below) 1. .7 + 0. ,3 16 + 6 1. .8 + 0. .3 4. .1 + 1, .0 2. .3 Phenol 0, .8 + 0. .3 3.3 + 1.0 0. .9 + 0. .3 1 . 4 + 0, .3 1, .5 N i 2 + 1, .7 + 0, .3 1, .5 + 0, .3 1, .4 + 0 .3 1 .0 Fe(CN)f" 2, .0 + 0 .5 2 .2 + 0 .4 2 .0 + 0 .4 1 .0 Naphthalene 0 .13 + 0 .03 13 + 2 0 .13 + 0 .03 0 .80 + 0 .05 6 .0 NaOSA is sodium octyl sulphate, whose c r i t i c a l micelle concentration is 0.14 M. Naphthalene data refers to sodium dodecyl sulphate (NaLS). b -4 Solute concentrations were -10 M. k M measured in 0.05 M NaOSA aqueous solutions, except for naphthalene, where [NaLS] = 0.003 M. ^ k M measured in 0.19 M NaOSA aqueous solutions, except for naphthalene, where [NaLS] = 0.016 M. Hexane for naphthalene, methanol for I 2 and phenol. -137-However, the observed increase above CMC is not as large as that obtained in a pure hydrocarbon solvent (column 2 in Table 7.1). This means that the inner micellar core is of a sufficiently ordered nature such that i t resem-bles a solid or very viscous environment [127], where reactions should be slower than that in a f l u i d hydrocarbon medium. A further important obser-vation is that k^ is not decreased by enclosure of the solute in a micelle. Although the micelle is a thick sheath, i t does not seem to impede Mu from reaching the solute in the micellar core. By analogy, this is similar to the efficient transmission of H atoms in radiobiological systems containing membranes of this type of composition. These results demonstrate the use-fulness of Mu as a non-destructive and sensitive probe in large macromolecular and biological systems. -138-7.B. Studies with hydrocarbons. Muon and muonium have been studied more extensively in water than in hydrocarbons. One d i f f i c u l t y with hydrocarbons lies in the reproducibility of purity in the sample. At times many attempts must be made before success can be realized for an experiment. This is especially relevant to MSR studies in these non-polar media. However, since the discovery of Mu atoms in various liquid hydrocarbons by Ito et a l . [33], many successful experi-ments [34,35,49,74,107] have been carried out in collaboration by groups : from TRIUMF and KEK. The ySR and MSR results of these works w i l l be high-lighted in this section with the emphasis on their implication for the model of Mu formation in condensed media. 7.B.I. Yields in liquid hydrocarbons. There are many interests in liquid hydrocarbons. By extrapolating the values of P^ , P^ , and A Q, in t r i n s i c properties and mode of formation of Mu can be investigated and compared for non-polar and polar media. Also, there is particular relevance to the studies of biologically important materials such as micelles and cyclodextrins, as already alluded to. The samples of hydrocarbons were a l l purified as described in chapter 2. The results are summarized in Tables 7.3 and 7.4. During the course of these experiments several samples of each liquid were studied following different purification procedures and with different rates of bubbling by helium gas in the target ce l l s . It was clear that effects of both organic impurities (probably carbonyl compounds and unsaturated hydrocarbons) and residual 0^ -139-c TABLE 7.3 Results obtained f o r Pp, P M, P L and X i n n-hexane, c-hexane, t e t r a -3. b methylsilane (TMS), methanol, and water ' . L i q u i d ! £ ! M ! L V 1 Q 6 5 - 1 n-Hexane 0.65 ± 0.02 0.13 ± 0.02 0.22 ± 0.04 1.0 ± 0.5 c-Hexane 0.69 ± 0.02 0.20 ± 0.02 0.11 ± 0.04 1.5 ± 0.4 TMS 0.53 ± 0.03 0.21 ± 0.02 0.26 ± 0.04 0.64 ± 0.07 Methanol 0.62 ± 0.01 0.23 ± 0.02 0.15 ± 0.02 2.3 ± 0.4 (0.74) d (0.61) (0.19) (0.20) (0.75) Water 0.62 ± 0.01 0.20 ± 0.01 0.18 ± 0.01 0.25 ± 0.08 d (0.62) (0.19) (0.19) (0.25) 3. Pp, P^ and P^ are the f r a c t i o n s of incident muons which are observed as diamagnetic species, free muonium atoms, or as neither of these (P. = 1 - Pp - Pj^), r e s p e c t i v e l y . X i s the observed muonium spin r e l a x a t i o n rate constant. k The values quoted are the mean of the values obtained independently from the l e f t - and right-hand detectors. The erro r l i m i t s given cover the two values and each of t h e i r s t a t i s t i c a l errors only. Calculated f o r neat methanol from 20% methanol i n water from Table 7.4. Mean values of the data published previously i n r e f s . [24, 29, 42, 43]. -140-TABLE 7.4 Results obtained for A in solutions (plus parameters calculated therefrom) Calculated Composition A / s - 1 A q ( s o l v e n t ) / s - 1 kM/M~1s"1 A o ( s o l u t e ) / s - 1 20% (vol) methanol 5.1 x 105 3.6 x 1 0 5 a 3 x 10 4 0.74 x 10 6 in water 1.9 x 10"4 M phenol 2.3 x 10 6 1.0 x 10 6 7(±2) x10 9 in n-hexane A A q in the sample of water used was considerably larger than is normal in pure water. -141-contribute to the X observed in a l l cases. It is especially important to bubble well since 0^ is about 100-times more soluble in hydrocarbons than in water. In general, X usually decreased somewhat when the gas bubbling -rate was increased. Therefore, i t should be emphasized that the results presented here correspond to the longest-lived Mu signals obtained in each liquid. However, this effect of C> did not affect P.. or P_. Also, once 2 M D the Mu decay constant X q has been determined for a neat solvent, i t was then used to make up solutions containing a solute to determine k^ (i.e. phenol in n-hexane in Table 7.4). There are two points of interests from these results. The f i r s t in-volves the intrinsic Mu reactivity in these solvents (i.e. X ). It can be seen from Table 7.3 that none of the non-polar media has smaller X than that of water (though neopentane now equals i t ) . Perhaps this suggests (neglec-ting 0^ and impurity effects) that these X s arises from a hydrogen abstrac-tion reaction by Mu. Even considering these effects, the reported X s for these non-polar media increase systematically in the series TMS/n-hexane/ c-hexane, as the -CH^ groups are replaced by groups. This trend is to be expected i f these X s arises from an H-abstraction reaction with Mu since the C-H bond strength of methylene is slightly less than that of methyl. In addition, the reactivity of Mu in hydrocarbons is demonstrated 9 -1 -1 by the result of phenol in n-hexane. The value of (7±2)xl0 M s in n-hexane is some 3-times faster than the analogous reaction in water [29] suggesting that the enhanced reactivity seen for phenol embedded in micelle [39] may be due to the change from water to hydrocarbon as the environment for the reaction. -142-The second interest from these results in liquid hydrocarbons concerns the Mu formation mechanism. On the basis of the spur model, one expects the value of Pp and P^ to depend on both the chemical composition of the medium -and their physical properties (such as dielectric constant and electron mobility). The comparison for these solutes are given in Table 7.5. It can be seen that P.., P_ and PT are essentially the same for both non-polar M D L hydrocarbons and highly polar hydroxylic liquids. This strongly suggests that the formation of Mu by intraspur neutralization is not a dominant process. For example, Table 7.5 shows that P M § P^ are essentially inde-pendent of electron escape probabilities in low LET spurs, and independent of the mobility of free electrons in these media. Such findings are contrary to the expectations of a spur model of muonium formation. 7.B.2. Temperature dependence of muonium in hydrocarbons. In chapter 5, yields in solid and liquid neopentane were presented. Here, an extended temperature dependence is given for c-hexane and n-hexane. The experimental details for these experiments are the same as those des-cribed in section 5.A.I. The results for c-hexane and n-hexane are displayed in Figures 7.1 and 7.2, respectively. For c-hexane, P^  could not be measured below -100°C due to a very large value of X^ below that temperature. That the value of X^ changes markedly at liquid/plastic-crystal, plastic/non-plastic(c-hexane), and liquid/solid transitions, indicates Mu is an excellent and sensitive probe to determine phase transitions for crystals. The increase in X^ at -143-TABLE 7.5 Comparison of P M and (1-Pp) with properties of the l i q u i d : ( i ) the f r a c t i o n of electrons which escape intraspur n e u t r a l i z a t i o n i n low LET r a d i o l y s i s ( G f i / G t ) ; ( i i ) the electron m o b i l i t y ( y ) b ; and ( i i i ) the s t a t i c d i e l e c t r i c constant (e). Liquid PM (1--v G f i , / G t y e Water 0. .20 0. .38 0. 62 0, .002 78 Methanol 0. .23 0, .38 0. 38 0 .0006 33 TMS 0, .21 0 .47 0. 16 100 1.8 c-Hexane 0, .20 0 .31 0. ,03 0 .4 1.8 n-Hexane 0 .13 0 .35 0. ,03 0 .08 1 .8 G f i i s the free ion y i e l d [103] and G t the t o t a l i o n i z a t i o n y i e l d taken to be 4.5 (±0.5) i n a l l cases. y i s i n c m 2 V _ 1 s _ 1 , data from A l l e n [102]. -144-0 -50 -100 -150 Temperature ( ° C ) Figure 7.1: Muonium and diamagnetic muon parameters f o r cyclohexane from room temperature down to -150 C. P^, and P^ are the muonium, diamagnetic muon and lo s t p o l a r i z a t i o n , r e s p e c t i v e l y . The l i n e s are drawn to a i d the eye. •145-3.0 "52.0 C o Si.o o 1.0 C o N O CL L - o -l iquid -50 -100 n» • -150 -50 -100 o -150 Temperature ( °C) Figure 7.2: Muonium and diamagnetic muon parameters f o r n-hexane from room temperature down to -150 C. The notations are the same as i n Figure 7.1. Note that there i s no s o l i d phase corresponding to p l a s t i c c r y s t a l s . -146-the solid phases can be explained in terms of increased dipole-dipole interactions. That i s , as a liquid these interactions are isotropic; however, at lower temperature, the random magnetic fields of surrounding --nuclear spins becomes anisotropic due to solidification with resulting interaction with the Mu dipole, thereby causing an increase in X^. A similar explanation can also be applied to the observed increase of X Q for c-hexane. For both c-hexane and n-hexane, the yields of P M and do not change sign i f i c i a n t l y in the solid phase. There is no disappearance of P^, as in ice [43]. These results resemble that in neopentane (Table 5.3). Again, the expanding track model can be used to rationalize P^ by postulating that there may not be any suitable trapping sites available for Mu in these hydrocarbons even at these low temperatures (see section 5.A.2.). 7.B.3. Effect of external electric fields on the ySR of liquid hydrocarbons and fused quartz. For positronium (Ps=bound state of e + and e ), i t s formation mechanism is commonly based upon a spur reaction model [128]. Indeed, the spur model of Mu formation [43] is a copy of that Ps mechanism. The spur model of Ps formation assumes that a positron thermalizes in i t s own terminal radiation spur, where i t can combine with one of the excess electrons to form a Ps atom. This model explains well the experimental results that good electron scavengers are also good inhibitors of Ps formation [129]. More relevant to this section, i t explains the fact that the application of an external -147-electric f i e l d (EEF) inhibits Ps formation efficiently [130], Therefore, an analogous experiment can be investigated for Mu to test the spur model. A special teflon c e l l was constructed for the experiment. Thin tungsten -meshed sheets (0.05 mm thick, 30 mesh), which were attached to the mylar films of a regular teflon c e l l , served as the electrodes. Less than 1 nA current was passing between the electrodes, therefore there were no side effects on the transverse magnetic f i e l d . For the experiments with the organic liquids, only Ap was measured. A^ was not attempted due to the d i f f i c u l t i e s mentioned in section 7.B.I. The presumption here was that i f the combination of y + and e occurs i t should be hindered by the EEF, thus more of the u + w i l l eventually precess as diamagnetic muon thereby increasing Ap. To study the effect of EEF on A^, fused quartz was chosen. Quartz is a suitable material since i t has a large value and the technical part of this particular experiment (as compared to an organic liquid) would not be d i f f i c u l t . In the organic liquid experiments up to 20 kV/cm was applied, while up to 60 kV/cm was used for fused quartz. Such fields should easily separate y from e in a typical low LET spur, sufficiently that an increase in Ap should be seen in accordance with the spur model. It can be seen clearly from Tables 7.6 and 7.7 that there were no changes observed for A Q or A^ ^ in a l l the liquids studied or for fused quartz, when an EEF was applied. Again, i t is entirely in accord with the hot model. Since the muon is in a charged state only when i t s energy is relatively high, the EEF w i l l have no effect on i t s charge-exchange processes. Furthermore, an EEF should not influence the hot reactions of Mu, or i t s thermalization. -148-TABLE 7.6 E f f e c t of EEF on A Q and Aj^ i n fused quartz. E (kV/cm) A D 0 0.0925 ± .0013 0.101 ± .001 15 0.0937 ± .0018 0.100 ± .001 30 0.0925 ± .0015 0.101 ± .001 45 0.0969 ± .0014 0.100 ± .001 -149-TABLE 7.7 Ef f e c t of EEF on Ap i n various l i q u i d s (the s t a t i s t i c a l errors are between 0.001 and 0.002 i n a l l cases). Left Right Liquid E = 0 E = 20 kV/cm E = 0 E = 20 kV/cm n-hexane 0.223 0.223 0.238 0.238 c-hexane 0.236 0.233 0.243 0.246 benzene 0.088 0.091 0.095 0.096 CS 2 0.105 0.105 0.121 0.122 CCX,. 0.338 0.334 4 -150-7.B.4. uSR studies with solvent mixtures. It has already been shown [4,28] that in a wide variety of common liquids varies from 0.17 (for benzene) to 1.0 (for CCS,^), while water and saturated hydrocarbons have P^ values of -0.63. There have been unsuccess-ful attempts to correlate these P^  data with various physical and chemical properties of the liquids [28]. There does not seem to be any consistent correlation with such properties as dipole moment, density, p o l a r i z i b i l i t y , dielectric constant or ionization energy. Some systematic trends were evident [28], however. For example, P^ has been found to be less than 0.5 only in molecules containing ir-bonds, and the greater the degree of conjugation the smaller is P^, with benzene being the extreme. Also, P^  increases with the number of halogen substitutions in organic compounds, with CC£^ being the upper extreme. It was thus decided to study solvent mixtures, in particular mixtures of c-hexane, benzene, and CCl^ and to determine P^ values to compare the hot and spur models of muon fates. Regular cells containing these solvent mixtures were placed in front of a surface muon beam to collect left and right histograms. The results are given in Figure 7.3 as plots of P^  against volume fraction of CC£^, or c-hexane, in benzene. The volume fraction is the best representation of these results since i t most closely reveals the fraction of time the muon spends in contact with each ingredient during i t s slowing-down process. Although these results cannot give any indications of why CCl^ and benzene should have the largest and smallest P^  values, they do however provide some information about the mechanism of Mu formation in liquids. -151-Figure 7.5: Plots of P^ against volume f r a c t i o n of CC£.^ or cyclohexane i n benzene (1) for CC£^-benzene mixtures and (2) for cyclohexane-benzene mixtures: ( o ) data points obtained on the left-hand side detector; ( A ) data from the right-hand side detector; (•) data taken from r e f . [28]. The dotted curve 3 i s expected on the basis of a spur model with CCZ^ as an intraspur electron scavenger, and curve 4 i s expected i f benzene protects cyclo-hexane by energy t r a n s f e r or s a c r i f i c i a l scavenging. -152-For CC£ 4, the fact that PD=1 can be explained, through the spur model, by the fact that CC£ 4 is an excellent electron scavenger, therefore giving CC£ 4 and then C£~+CC£3, which are long-lived. Thus an ion, rather than an electron, would be available for neutralization by y +, as in (7.2), y + + C£~ (or CC£ 4") > MuC£ (or CCi^. (7.2) This neutralization, due to CC£ 4 s low dielectric constant, would occur over considerable distances and therefore up to times when the original spur has virtually dispersed due to diffusion. But, i f such intraspur reac-tions are responsible for Pp=1.0 in CC£ 4, then they should also be equally effective when the spurs are partly composed of benzene molecules (e.g. 0.5 mole fraction CC£ 4 in benzene) because CC&4 would equally scavenge a l l electrons i n i t i a l l y . Thus, on the basis of the spur model, one should have obtained a pronounced curve, such as that sketched as curve 3 in Figure 7.3. The fact that the experimental data points (curve 2) for these CC£4~benzene mixtures is linear rather than following curve 3 indicates once again that the spur process plays a minor role in the Mu formation mechanism. A similar argument can be applied to the benzene-cyclohexane mixtures. The delocalized ir-bonding of benzene should be effective in reducing P Q of I I it cyclohexane when they are mixed together. If a sponge-like -protection by benzene is operative [131], then one should have obtained a curve such as 4 in Figure 7.3. The disagreement between curve 2 f} 4 and the fact that -153-curve 2 is approximately linear implies that there is a one-step intra-molecular process for these mixtures. These results are not inconsistent with the expectation of a hot model of Mu formation. 7.C. Muonium solution kinetics. Since the discovery of Mu in water by Percival et al.[24], many reac-tions of Mu interacting with various solutes in aqueous solutions have been studied by groups from TRIUMF [29,30,70] and SIN [87]. Chapter 6 detailed the kinetics of Mu reacting with DCO^  and OH . In this section, more recent results including Mu reactions with vinyl monomers [41,93], nickel cyclam [40], and cyanides [123] w i l l be br i e f l y described. The discussion w i l l be mainly focused upon kinetic isotope effects (k^/k^), i t s relevance to H atom studies, applications for polymer i n i t i a t i o n rates, and the use of Mu as a sensitive probe for paramagnetism in solutions. The MSR tech-nique was used for a l l these experiments, including the temperature-dependence measurements. Mainly a surface muon set-up was used, with backward muons on occassions. A l l quoted values of k^ have r e a l i s t i c probable errors of ±25%. -154-7.C.I. Muonium addition to vinyl monomers. As already seen in chapter 4, Mu radicals can be observed in benzene, styrene, and their mixtures. These species are formed by Mu addition reac-tions. This section describes Mu additions to aqueous solutions of styrene, methylmethacrylate (MMA), acrylonitrile (AN), acrylic acid (AA), and acryl-amide (AM). Also, MRSR experiments were carried out to obtain A^ in order to identify the mode of reaction. The A -value for styrene has already been shown to be independent of magnetic f i e l d (Figure 2.6). By comparing i t s value of 214 MHz to analogous radicals observed by ESR, one identifies the mechanism as Mu addition to the vinyl bond of styrene (see chapter 4). This contradicts the suggestion i by Swallow [132], based on-band labelling studies, that H adds to styrene s vinyl bond 15% of the time and to the ring 85% of the time. However, one must realize that the radical adducts observable in MRSR are in the time - 7 - 5 window of 10 to 10 sec, therefore i t is quite likely that the observed radicals are not the primary ones but rather the most stable ones after intramolecular rearrangements. This same conclusion has also been reached during the discussion section in chapter 4. Table 7.8 l i s t s the hyperfine coupling constants for the various mono-mers in neat solutions. With the exception of MMA, a l l monomers gave one observable Mu radical. The comparison with A^ confirms the assumption that Mu is indeed adding across the vinyl bond of these monomers as shown here by equation (7.3): -155-TABLE 7.8 Hyperfine Coupling Constants . Monomer Mu-R H-R A /MH_ A'/MHz A /MHz y 2 _y _J> Styrene 213.4 67.0 56.0 MMA 270 84.8 73 276 87.0 71.2 AN 280.4 88.1 64.4 AA 319.3 100.3 70.8 Benzene 514.6 161.6 133.7 -156-R 1 R 1 Mu + C = CH„ > CH9Mu (7.3) / 1 „ / 2 R2 R2 where Rj and R2 vary through H, CONH2, C02H, CH^, CN d, C0 2CH 3 depending on the monomer. In the case of MMA, the two radicals have been assigned as the cis and trans forms where Mu adds at the CH2~group end of the carbon-carbon double bond. This is analogous to the assignment by Roduner et a l . [133] for ethylmethacrylate. As one can clearly see, the radical assign-ments in Table 7.8 identifies the Mu reactions with these monomers as that of an addition reaction. Table 7.9 l i s t s the reaction rate constants (k^) for the various mono-mers. Maleic acid is included for comparison as a non-acrylic compound. Due to the scarcity of H atom data, k^/k^ ratios were obtained for only AN, i AM, and maleic acid. With the exception of styrene, a l l k^ s are near the diffusion controlled limit (-2x10*^ M * ) . The low value of k^ for styrene 9 -1 -1 (1.1x10 M s ) indicates that the large n-delocalization stabilizes the molecule and hence reduces i t s reactivity toward Mu. For the other monomers, 9 -1 -1 k^ increases from 9.5x10 M s in the order MMA<AN<AA<AM. This trend can be explained on a steric argument. For example, MMA should have the smallest It M k^ since i t has the least accessible double bond due to the blocking effect of i t s a-methyl group. For the three straight chain 3-carbon conjugated monomers, Mu (which acts like a nucleophile) should encounter less electron shielding and react faster with AM or AA due to the electron withdrawing -157-TABLE 7.9 Reaction rate data of monomers at 295 K . Monomer k M/10 1 0M - 1s 1 kM / kH + Styrene 0.11 ± 0.01 Methylmethacrylate 0.95 ± 0.19 A c r y l o n i t r i l e 1.14 ± 0.20 2.8 A c r y l i c acid 1.55 ± 0.25 Acrylamide 1.90 ± 0.13 1.1 Maleic acid 1.1 ± 0.1 1.4 k„ obtained from reference [99], data only available H f o r AN, AM and maleic a c i d . Maleic acid data was added as comparison. -158-CO group on these molecules; whereas i t would be slower with AN. When comparing with k^, AM s isotope effect of 1.110.4 indicates that there is l i t t l e mass effect in this type of reaction. As these reactions are near the diffusion controlled limit, the question arises as to whether the Stokes-Einstein relationship holds for small particles like Mu or H, or whether the isotope effect should be 3, as with AN, because the mean velocity of Mu (with mass 1/9 H) is three times faster than that of H. At this stage, due to the lack of H atom data, judgement is reserved on the kinetic effect in these addition reactions. That the H atom data is of such scarcity indicates the strong need for the Mu chemist to supply information on the ini t i a t i o n of polymerization of monomers by the simplest of atoms, the hydrogen atom. 7.C.2. Spin conversion reaction of muonium with nickel (II) cyclam. The f i r s t observation of electron-spin exchange (or spin f l i p ) inter-actions of Mu was by Hughes in 1960 s [10] studying Mu reactions with and NO in the gas phase. Jean et al.[30] have investigated interactions of various paramagnetic ions and found that Mu reacted with a low-spin complex (diamagnetic Fe(CN) 4 ) at -10 8 M \ while in a high-spin state (para-magnetic Fe 2 +) i t reacted at -10"^ M *s In this section w i l l be described 6 aq the reaction of Mu with a metal complex (nickel (II) cyclam) in two spin states, interchanged merely by altering the ionic strength of the solution by addi-tion of an inert salt. Nickel cyclam, (1,4,8,11-tetraazacyclotetradecane)-nickel(II), is an excellent system for such a study since i t switches from be-g ing an octahedral (paramagnetic, d ) configuration in aqueous ammonia to a square-planar (diamagnetic) structure in solution containing perchlorate or sulphate. -159-In the equilibrium between the paramagnetic (P) and diamagnetic (D) species of Ni-cyclam, Mu reacts with each with different s, represented by: kP Mu + P > products (7.4a) kD Mu + D » products. (7.4b) The overall k Q b s can be equated with k p[P]+k D[D], which leads to the follow-ing relationship [40]: ( X - X Q ) = k p[Ni]F p + k D [ N i ] ( l - F p ) , (7.5) where F p is the fraction of complex ions in the paramagnetic state. There-fore, a plot of (X-X Q)/[Ni] versus F p, shown by Figure 7.4, should give k Q as i t s intercept and (kp-k^) as the slope. Unfortunately, Figure 7.4 does not give a straight line. Nevertheless, 8 -1 -1 the intercept at Fp=0 from the three low F p points gives k^<5xl0 M" S " , whi seems to be a reasonable value, being some 100-times slower than the diffus-ion controlled limit (as with Fe(II), for instance), k^ can be a result of a reduction reaction (Ni (II)->-Ni (I)) of the diamagnetic product, of which the analogous H atom reaction has a value of 2x10^ M *s 1 [99], It could also be a reaction between Mu and the ligand (abstraction of H) since the blank 8 -1 -1 solution of the cyclam ligand also gives a of less than 5x10 M s -160-0.0 0.2 0.5 0.7 1.0 F P Figure 7.4: Plot of the observed rate constant, given as (X-X o)/[Ni], as a function of the f r a c t i o n of nickel-complex species i n the para-magnetic state (Fp). [Ni] i s the t o t a l Ni complex i n s o l u t i o n . The two highest data points were obtained i n aqueous NH^ s o l u t i o n , where the a x i a l ligands w i l l be NH^, whereas the lower three points correspond to H^ O ligands i n the a x i a l p o s i t i o n of the paramagnetic component. -161-An intriguing question is the non-linearity of Figure 7.4 at the high Fp data points. In any case, i t is clear that k p is much greater than k D > Using the i n i t i a l slope at the three low F p-points, one obtains kp=4.5x10"^ -M *s Two reasons can be invoked to explain the non-linearity. The f i r s t arises from the fact that NH^  had to be added to induce the change to a paramagnetic state for the two. high F p data points. The second could be a result of the presence of a five-coordinate intermediate, which, like penta-cyanonickelate [134], could be diamagnetic. In any event, the results imply kp for the interaction of Mu with the paramagnetic complex to be -4.5x10^ M *s * when H^O is the axial ligand and -2x10"*"^  M *s * for NH^  as the axial ligand. This experiment further demonstrates the potential of using the i i I I I I i t t r i p l e t to singlet conversion of muonium as a probe of paramagnetism and as a monitor of spin-exchange interactions. 7.C.3. Muonium addition to cyanides. In section 7.C.I., the observed isotope effects of 1.1 and 2.8 for acrylamide and acrylonitrile prompted more extensive investigations into Mu addition reactions. Here, both organic and inorganic cyanides were selected for study. Similar to ESR studies [135,136], one expects Mu to add across the C=N bond of these cyanide molecules. An MSR technique employing surface muons was used to extract the k^ s; also a temperature-dependence study was carried out for cyanoacetate and tetracyanocadmate(II) 2-(Cd(CN)4 ) to determine their Arrhenius parameters. -162-Table 7.10 l i s t s the kinetic data for these cyanides and other relevant i Mu rate constants. By applying Smoluchowski s equation (1.10) (see also 2-chapter 6) for the cyanoacetate and Cd(CN)^ reactions, i t was found that 2-Cd(CN)^ exhibited diffusion controlled behaviour while cyanoacetate demon-strated activation controlled behaviour. This was reflected by the activa-tion energies of 35 and 15 kJ mole 1 [70] obtained for cyanoacetate and 2_ Cd(CN)^ , respectively. Because the C E N bond system is stronger than the C=C double bond of the monomers, with more electron density to shield the nucleophilic Mu from the target C atom, one expects the cyanide rate con-stants to be smaller than those for the monomers (Table 7.9). Indeed, 1 7 10 -1 -1 Table 7.10 gives the cyanide k^ s ranging from 5.1x10 to 1.7x10 M s These data suggest that the reactions of Mu with organic n i t r i l e s and cyanide to be activation controlled while i t is diffusion controlled for 2- 2-Cd(CN)^ . The peculiarity of the Cd(CN)^ can possibly be explained by a size effect increasing the col l i s i o n cross-section and making available four CN groups on each encounter. Another possibility is that the cadmium complex stabilizes the radical product to a much higher degree than the other cyanide reactants, thereby lowering the reaction barrier. Also listed in Table 7.10 are the kinetic isotope effects, k^/k^ for the solutes studied. That k^/k^ has values of 0.7 and <7 for the inorganic cyanides supports the idea that addition reactions, being primarily elec-tronic in nature, should have l i t t l e or no mass effect, as exemplified by the monomer experiments in section 7.C.I. However, both the organic n i t r i l e s react much faster with Mu than H giving k M/k of 24 for cyanoacetate and 19 -163-TABLE 7.10 Ki n e t i c data of cyanide system with some associated Mu rate constants. Solute k M / M " l s _ 1 Ea/kJ mole 1 V k i cyanoacetate 7.7 x 10 7 35 ± 2 24 a c e t o n i t r i l e 5.1 x 10 7 19 cyanide 3.0 x 10 9 0.7 tetracyanocadmate(II) 1.7 x 1 0 1 0 15 ± 2 <7a a c e t i c a c i d ~3.8 x 10 6 cadmium(II) 8.5 x 10 5 H a k was reported i n reference [99] as >2.4 x 10 M s . -164-for acetonitrile. Percival et al. [43] have found a similar isotope effect of 40±10 for reaction of Mu with acetone in water, the only other organic addition reaction studied to date. These data, rather than clarifying the -questions of Mu addition isotope effects, reflects the necessity for more kinetic information; particularly activation controlled addition reactions with organic solutes. -165-CHAPTER 8 SUMMARY AND CONCLUSION -166-This Ph.D. t h e s i s i s , i n a sense, a continuation - a more in-depth i n v e s t i g a t i o n - of muonium chemistry r e s u l t i n g from the M.Sc. t h e s i s . Therefore i t seems appropriate to now present a b r i e f summary of a l l work -during my graduate years at U.B.C. and TRIUMF. From the magnetic f i e l d dependence study i n chapter 3 and the temperature-dependence of X Q from the M.Sc. work, i t has now been v e r i f i e d that Mu does not react with water to any s i g n i f i c a n t extent. Instead, the natural r e l a x a t i o n of Mu i n water (A ) i s simply an experimental a r t i f a c t , due to a combination of magnetic f i e l d inhomogeneities and the physical phenomenon of Mu-frequency beating. Perhaps the use of the two-frequency MSR technique can be applied to study the o r i g i n of A q i n other solvents. For the f i r s t time, muonium r a d i c a l s were observed and i d e n t i f i e d through t h e i r combined MRSR precession frequencies i n high transverse magnetic f i e l d s i n mixtures. Thus, pure benzene, pure styrene and a f u l l range of t h e i r mixtures were studied, a l l as l i q u i d s at room temperature. In styrene, i t was found that there was only one r a d i c a l formed, one r e s u l t i n g from Mu addition to the v i n y l bond of styrene, while there was no net i n s e r t i o n of Mu into the phenyl r i n g as i n benzene. In benzene-styrene mixtures, the r a d i c a l s obtained -9 -5 ' i n each pure l i q u i d were both present, so no slow (10 - 10 sec) intermole-cular exchange occurs; but strong s e l e c t i v i t y was found with the formation of the r a d i c a l from styrene being about eight-times more probable than the r a d i c a l from benzene. Further, the s e l e c t i v i t y r e s u l t from chapter 4 implies e i t h e r that the r a d i c a l precursor i s epithermal, or that the e f f e c t i s due to the t u n n e l l i n g a b i l i t y of the l i g h t Mu atom, or there i s intramolecular rearrange-ment . -167-To s p e c i f i c a l l y investigate the hot and spur Mu formation mechanisms, muon and muonium y i e l d s i n neopentane ( l i q u i d 5 s o l i d ) and concentrated OH solutions were studied. F i r s t , the observed long-lived Mu signal i n neopentane fur t h e r substantiated the conclusion reached i n chapter 3, that A q i n pure, unreactive, solvents i s a physical a r t i f a c t rather than a chemical reacti o n . Second, both and P^ r e s u l t s i n neopentane and concentrated OH solutions seem to be at variance with the expectations of a spur model of Mu formation. In muonium solu t i o n k i n e t i c s , many reactions were investigated. In the M.Sc. th e s i s , i t was shown from temperature dependence studies of k^ that Mu reacted at d i f f u s i o n c o n t r o l l e d rates with MnO^ , maleic a c i d and N i 2 + (E ~ 17 kJ mole * ) , and at an a c t i v a t i o n c o n t r o l l e d rate with HCO_ (E ~ 33 kJ m o l e - 1 ) . On the other hand, the low E (8 kJ mole *) obtained cl 2-f o r NOg was explained v i a quantum mechanical t u n n e l l i n g contributions or a l t e r n a t i v e (competitive) reaction paths. In t h i s Ph.D. t h e s i s , the reaction of Mu with OH was found to be r e l a t i v e l y slow due to a substantial a c t i v a t i o n aq J energy (E ~ 40 kJ mole . When compared to H atom r e s u l t , there was no k i n e t i c isotope e f f e c t at room temperature. The reaction was i d e n t i f i e d as a c t i v a t i o n c o n t r o l l e d and i t was concluded that Mu behaves as a "muonic a c i d " i n t h i s acid-base t r a n s f e r r e a c t i o n . As with the OH reaction, the abstraction of Mu with DC0_ gave a large E of 39 kJ mole 1 as i f i t i s p r i m a r i l y 2, a a c t i v a t i o n - l i m i t e d . When compared with HCO,,, the isotope e f f e c t s (k^/k^ and k H/kp) implied that quantum mechanical t u n n e l l i n g does not dominate the abstraction of H and D atoms i n HC02~ and DC02" by e i t h e r H or Mu. Further-more, c o l l a b o r a t i v e works (section 7.C.) describing reactions of Mu with -168-cyanides, vinyl monomers, and nickel cyclam have shown that there is almost no mass-effect in addition reactions. But, more importantly, these solution kinetics results demonstrated the potential of using Mu in polymer in i t i a t i o n kinetics due to i t s unique characteristic as a sensitive and non-destructive probe. It is hoped that these kinetic studies w i l l be extended to non-polar (hydrocarbon) solutions and hence to a wider temper-ature range. In this way the effect of quantum mechanical tunnelling can be more easily manifest, as in either Arrhenius curvatures or relative primary isotope effects. In the M.Sc. thesis, the study of Mu reaction with C^faq) showed that the reaction did not involve any long range encounter. Unfortunately, the nature of the reaction, spin conversion or chemical reduction, could not be determined with the transverse f i e l d technique. However, the study did demonstrated the importance of residual 0^ in muonium kinetics. Since H atoms are involved in some of the most fundamental processes in biological systems, the MSR technique was also applied to study a few model biological systems. The macromolecules investigated were porphyrins (M.Sc), cyclo-dextrins and micelles (Ph.D.). The results again demonstrated the potentially sensitive nature of the muonium atom as a nuclear probe. Hopefully, this w i l l inspire future practical applications of this novel technique to other biological systems. Last, but not least, hydrocarbons were investigated to expand the study of Mu beyond aqueous media. These systems included various neat hydrocarbons, with studies of Xn and various muon yields, their temperature dependence, -169-the e f f e c t of EEF, and muon y i e l d s i n so l v e n t mixtures. A l l of the r e s u l t s tend to poi n t toward the f a c t t h a t Mu formation does not occur p r i m a r i l y v i a spur processes. Rather, the r e s u l t s are co n s i s t e n t w i t h those expected -to r e s u l t from a 'hot' mechanism. A l s o , the use of an 'expanding-track' model e x p l a i n s the r e s u l t s o f i n p o l a r media and at phase boundaries. Almost a l l r e s u l t s i n both the M.Sc. and Ph.D. t h e s i s seem to be i n c o n f l i c t with the spur model of Mu formation. I t i s i n t e r e s t i n g that the analogous model i n positronium s t u d i e s [129] i s now ga i n i n g wide acceptance. Then, one should ask, why does a spur model apply to Ps but not to Mu? This can be explained by r e a l i z i n g that there are s e v e r a l important d i f f e r e n c e s between the muon and p o s i t r o n at the end of t h e i r r a d i a t i o n t r a c k s . F i r s t , according to estimates by Mozumder [45], the l i n e a r energy t r a n s f e r (LET) of the p o s i t r o n g r e a t l y exceeds that of the muon (-two orders of magnitude) during t h e i r l a s t few hundred eV of k i n e t i c energy. Therefore, i t i s much more reasonable f o r the e + to create i t s own t e r m i n a l spur r e l a t i v e to the muon. Indeed, the l a s t spur of the muon i s a c t u a l l y created by a secondary e l e c t r o n , and i t i s d i f f i c u l t to envisage the muon donating a l l of i t s remaining k i n e t i c energy i n one l a s t i o n i z a t i o n process. This i s probably why an expanding track model [51] can more r e a d i l y e x p l a i n the missing f r a c t i o n s i n va r i o u s l i q u i d s . A second d i f f e r e n c e between Ps and Mu spur processes o r i g i n a t e s from the i o n i z a t i o n energy (Ip) °£ these two sp e c i e s . The 1^ of Ps (6.8 eV) i s l e s s than that of the molecules of a t y p i c a l medium (eg. I - 9-12 eV), whereas that of Mu i s higher (I p(Mu) = 13.6 eV), t h e r e f o r e one would expect the muon to continue e x t r a c t i n g e l e c t r o n s from the medium -170-down to thermal energies; however, the positron cannot undergo the capture step of such charge-exchange cycles during i t s last few eV of energy. A third factor concerns the relative thermalization distance of e + and y + after formation of their terminal spur. However, i t is not so clear whether or not this favours intraspur Ps formation over Mu [51]. With a l l these arguments and conflicting results, based on radiation chemistry effects, the spur model of Mu formation [43] should be seriously revised. Yet i t is un-fortunate that the hot model of Mu formation is so simple and vague that i t cannot be tested as severely as is the case with the spur model. At present, the muon community is rapidly expanding into many new areas of research, even into emission spectroscopic studies [3]. For certain, Mu w i l l continue to be useful as an isotope of hydrogen until the complicated picture of H atom chemistry is sorted out. It is hopeful that one can someday evaluate the pKa of Mu and to determine its standard reduction potential, to observe i t s tunnelling at low temperatures, to study i t s precession in hydrogen-bonded structures, to evaluate the solvation energy of y +, and to reach a consensus on the mechanism of Mu formation. To carry out such research at TRIUMF, the present apparatus, electronics, and counter set-up must be improved. For example, effects such as unstable c e l l geometry, magnetic f i e l d inhomogeneity, small positron solid angle detection, and electronic resolution a l l contribute to uncertainties in ySR, MSR and MRSR measurements. 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Swallow, Adv. i n Chem. Ser., 82_, 499 (1968). 133. E. Roduner, W. Strub, P. Burkhard, J . Hochmann, P.W. P e r c i v a l , H. Fischer, M. Ramos and B.C. Webster, Chem. Phys., 67, 275 (1982). 134. R.A. Penneman, R. Bain, G. B i l b e r t , C.H. Jones, R.S. Nyholm and G.K.N. Reddy, J . Chem. S o c , 2266 (1963). 135. P. Neta and R.W. Fessenden, J . Phys. Chem., 74, 3362 (1970). 136. D. Behar and R.W. Fessenden, J . Phys. Chem., 76^ , 3945 (1972). 137. P.W. P e r c i v a l and H. Fischer, Chem. Phys., 16_, 89 (1976). • 1 8 0 -APPENDIX I CORRECTIONS FOR MU-RADICAL AMPLITUDES -181-For uSR and MSR experiments, i t i s s u f f i c i e n t to obtain Pp's and P^'s by normalizing against CCl^ as given by equations (2.11) and (2.12). However, f o r MRSR experiments, since the observed r a d i c a l frequencies are so much higher than those i n the above techniques (at le a s t 10 times f a s t e r o s c i l l a -t i o n s ) , there arises a need f o r corrections of the observable amplitudes before normalizing against CCl^. The two corrections are, the f i n i t e response time (timing resolution) of the detection system and the pre-set number of bins per cycle i n the spectrum. Both of these can cause apparent reductions i n the amplitudes that one i s t r y i n g to observe. Depending on the l i m i t a t i o n s of these two e f f e c t s , amplitudes at frequencies greater than 200 MHz can be dramatically reduced or s e r i o u s l y obscured. I.a. E f f e c t of timing r e s o l u t i o n of the detector system on the amplitude of  sin u s o i d a l o s c i l l a t i o n s . Consider a wave of type cos (ait) (or s i n (art)) and that at t=0 i t fluctuates with a Gaussian d i s t r i b u t i o n of standard deviation T. This represents the detector response time (or the l i m i t of the measuring instrument) and i s commonly c a l l e d the " j i t t e r " i n one's system. This j i t t e r i s assumed to be Gaussian since most random noise phenomena are of that nature. This Guassian d i s t r i b u t i o n i s displayed by Figure I - l and i s represented by the following expression: f ( t ) = A . exp(-t 2/2x 2) ( I - l ) oo where A i s a fa c t o r so that f ( t ) i s normalized to 1 ( i . e . / f ( t ) d t = l ) . -182-co c o 7 0 0 6 0 0 -5 0 0 -4 0 0 -3 0 0 -2 0 0 -100 0 2 0 3 0 4 0 5 0 6 0 7 0 C h a n n e l n u m b e r Figure 1-1: Timing r e s o l u t i o n curve of Positronium system by measuring the two y's emitted from a 6°Co source. The curve i s Gaussian in nature and i s represented by equation (1-1). -183-As a matter o£ convenience, 2T is defined as the timing resolution (At) of one's system in this thesis. Notice that the value of f u l l width half maximum (FWHM) is slightly greater than that of 2 T. To determine the effect of how this j i t t e r , equation (1-1), affects the amplitude of a sinusoidal oscillation, one can physically superpose this j i t t e r on top of an oscillation and watch its effects as i t moves along the sinusoidal wave. Mathematically, this i s equivalent to doing a convolution of the two functions. Therefore, let's define a sinusoidal oscillation by the following expression, To perform the convolution, one integrates f(t') on S(t) over dt' from -°° to S(t) = cos(a)t) (1-2) CO CO I(t) = / S(t + f ) f ( f ) dt' ; (1-3) — CO or • 2/2x 2) d f . (1-4) — oo The integration can be easily carried out to yield, 2 2 - T 10 —oo (1-5) Notice the integral in wiggly brackets is simply the original Gaussian j i t t e r f(t) except with a phase shift in time. -184-Since £(t) and £(t' +ix uO are both normalized to one as mentioned previously, the e f f e c t I(t) i s again j u s t a simple Gaussian function with an 2 2 exponential decay of exp(-i u> /2) . For example, where u = 2TTV, and v (=1/T) i s one's observable frequency i n the J JSR, M S R or M R S R experiment. Notice that the d e r i v a t i o n i s the same f o r a sine function. 2 2 This observable amplitude f r a c t i o n , I"(t) = exp{-2ir ( T / T ) }, i s plotted against 2T / T i n Figure 1-2. It can be c l e a r l y seen that as T approaches At, i t w i l l not be observable i n the Fourier transform ( i . e . at 2x/T=l, l"(t)=0). This means that the following r e l a t i o n s h i p should be seen f o r amplitudes of various frequencies There are two ways in which one can check or obtain equation (1-7) . One i s to simply produce a timing r e s o l u t i o n curve of the system using muons and beam positrons. This i s done by using the muon as the s t a r t pulse and the beam e + as the stop pulse. The two signals are fed into a high r e s o l u t i o n multi-channel analyzer ( M C A ) , and a " T Q " curve can be obtained. This was ca r r i e d out f o r the backward muon set-up with the forward and perpendicular histograms. The forward-T^-calibration curve i s shown i n Figure 1-3. Since 2 t h i s i s a Gaussian d i s t r i b u t i o n , one can plo t In N(t) versus t to obtain At. In t h i s case, At =0.80 nsec (FWHM = 0.94 nsec) was obtained f o r the forward I(t ) = exp(-x a) /2) cos(ut), (1-6) •185-TD E < c o - t— 1 O D _0) J D D > CD CO _Q O 0.0 0.0 0.5 1.0 1.5 2T/T (Time r e s o l u t i o n / P e r i o d ) Figure 1-2: The e f f e c t of timing r e s o l u t i o n (2T or At) on a sinusoidal i s p l o t t e d against 2x/T 2 2 function. The observable f r a c t i o n a l amplitude exp{ - 2 t r ( T / T ) } -186-00 c O o 2680 -670 h 100 125 150 175 200 225 Channe l n u m b e r Figure 1-3: ^ - c a l i b r a t i o n or timing r e s o l u t i o n curve of the forward histogram f or backward muon set-up. -187-histogram, while At = 0.97 nsec (FWHM = 1.14 nsec) f o r the perpendicular histogram. It was experimentally impossible to obtain the At-value f o r the backward spectrum. However, t h i s T^-method of getting At i s only approximate "Since the clock of the detection system was not included in the c a l i b r a t i o n . Another method of obtaining the r e l a t i o n s h i p between the dependence of amplitude on At i s by emp i r i c a l l y measuring A^ of quartz as a function of magnetic f i e l d . Since A^ of quartz i s large (P - 0.80), then i n e f f e c t , I"(t) can be e a s i l y determined as a function of frequency. This i s a more accurate method compared to the above T^-MCA technique since the exact e l e c t r o n i c l o g i c , experimental set-up, and computer software are being used to obtain the A^ functional dependence on magnetic f i e l d . The timing r e s o l u t i o n curve i s given i n Figure 1-4. When f i t t e d with equation (1-7), ( i . e . In ^ ( 0 b s ) / % j ( r e a i 2 versus (freq.) ), At value of 1.24 nsec was obtained f o r the forward histogram, while 1.31 nsec f o r both the perpendicular and backward histograms. This shows that the j i t t e r i s Gaussian and the reduction e f f e c t can be described approximately by equation (1-7) . However, f o r the correction due to timing r e s o l u t i o n , the l i n e drawn i n Figure 1-4 was used to obtain the true A^ values f o r the r a d i c a l s . This i s shown i n Table 4.2. I.b. E f f e c t of packing f a c t o r on amplitudes. Due to the l i m i t a t i o n imposed by the clock (1GHz o s c i l l a t o r , 1 ns/bin absolute accuracy), the e f f e c t of packing f a c t o r (number of nsec per bin) on amplitude signal was investigated. The Mu-signal i n water at 9G (raw packing f a c t o r = 2 ns/bin) was f i t t e d with various packing f a c t o r s (up to 68 ns/bin) using equation (2.6). The e f f e c t i s plotted in Figure 1-5. It can be seen -188-0 100 200 300 400 500 Frequency/MHz Figure 1-4: Timing r e s o l u t i o n curve of backward muon set-up using quartz as the emperical measurement. Its observable i s plotted against frequency (v)• The s o l i d l i n e i s drawn by eye. The points are averages of the forward, perpendicular and backward histograms. -189-0.0 0.2 0.4 0.6 0.8 1.0 1.2 2T/T ( b i n s / M u — per iod) Figure 1-5: E f f e c t of b i n s / o s c i l l a t i o n period on amplitude. The observable Mu-amplitude at 9G i s pl o t t e d against 2T/T (bins/Mu-period). -190-that the functional dependence resembles that of equation (1-7) . As a matter of f a c t , considering s t a t i s t i c a l e f f e c t s at high packing f a c t o r s , the d i s t r i -bution i s Gaussian from 100% A^ down to 50%. For MRSR experiments, Ins/bin packing was used f o r the r a d i c a l spectra. Therefore, Figure 1-5 i s used as a further ( i n addition to I.a.) small c o r r e c t i o n to the amplitudes at various frequencies. This i s given i n Table 4.2. I.e. Other e f f e c t s . Besides the above two corrections, a d d i t i o n a l e f f e c t s such as density of sample and muon stops i n target walls should be considered. The f i r s t e f f e c t a r i s e s from the f a c t that higher-energy decay positrons contain "higher asymmetries" of the t o t a l muon p o l a r i z a t i o n than lower-energy positrons [4]. This means that high density materials (by degrading low energy positrons more) therefore create an apparently l a r g e r asymmmetry r e l a t i v e to low density materials. The second e f f e c t depends on col l i m a t e r s i z e up to the sample container, i n t h i s case, pyrex glass. In t h i s t h e s i s , backward muons are collimated to 20 mm and i t s t h e i r energy adjusted by a water degrader so that the muons w i l l stop i n the middle of the sample; therefore muons w i l l only stop i n the front glass of the sample container. It i s estimated that these two e f f e c t s , density and muon stops i n glass, are very small f o r the MRSR experiments with benzene and styrene. In addition, the two e f f e c t s tend to cancel each other out. This i s corroborated by the f a c t that pSR experiments using CC£ 4, H 20, and the mixtures gave the same P D values f o r a l l three histograms. This indicates that c o r r e c t i o n i s not needed f o r these two e f f e c t s . -191-APPENDIX II RESIDUAL POLARIZATION OF A SINGLE-STEP MU REACTION MECHANISM -192-In section 5.B.2., an expression f o r the expected P^-values f o r a s i n g l e -step Mu reaction mechanism was given as equation (5.9). The de r i v a t i o n leading up to that expression was based upon some c l a s s i c a l and i n t u i t i v e "arguments. In t h i s appendix, a formal d e r i v a t i o n based on the formalism by Fleming et a l . [28] w i l l be presented. Other references [4,137] should be read by those who wish to f u l l y understand the complete d e r i v a t i o n of muon p o l a r i z a t i o n and i t s time dependence i n transverse magnetic f i e l d s . II.a. Residual p o l a r i z a t i o n i n l i q u i d s . The r e s i d u a l p o l a r i z a t i o n ( P r e s ) i s defined as the amplitude and phase of the diamagnetic y +-molecule extrapolated back to t=0, the time at which Mu i s i n i t i a l l y formed. The observable signal (to a good approximation) i s the r e a l part of P(t) [75], the exact (complex) time dependent p o l a r i z a t i o n of the e n t i r e muon ensemble. Since an observation on the ySR timescale takes place long a f t e r the rapid motions of the y + - s p i n i n Mu have s e t t l e d down to the slow precession of the free y + - s p i n i n a diamagnetic environment, one can say that one observes at t=°°. For example, P Q B S ( t ) = lim (Re) P(t) . ( I I - l ) t-x» Experimentally, the signal i s f i t t e d to an expression of the form, P O B S ( t ) = lPresl C o s ( V + •> ' ( H " 2 a ) i u D t or P 0 B S ( t ) - Re { P r e s • e } ; (II-2b) - 1 9 3 -where tan<j> = {Im(P )/Re(P_._)}. By equating ( I I - l ) and (II-2b), one has P = lim P(t) • exp(-icj Dt) . (H-3) t-*» Notice here that P(t) depends on the method by which Mu reacts. In t h i s case, the following derivations w i l l be f o r a s i n g l e step Mu reaction mechanism. II.b. Implication of Mu reaction k i n e t i c s on P r e s • Since one e s s e n t i a l l y receives one muon at a time [3] into a sample, the problem of Mu k i n e t i c s , measured one at a time, can be equated with a "steady-state" approximation. Also, one can assume pseudo f i r s t - o r d e r k i n e t i c s i n t r e a t i n g the Mu decay constant X as proportional to concentration S ([S]) : thus, X = k[S] + A Q . (H-4) There are two obvious l i m i t s f o r X i n r e l a t i o n to P r e s - (i) If ^ > > a ) 0> t h e muon spins w i l l not have moved appreciably before the Mu reacts, then there w i l l be no dep o l a r i z a t i o n of the muon spins. This gives P = 1 . (H-5) res ( i i ) I f A<«JI>m, then the reaction at randomly d i s t r i b u t e d times w i l l leave the muons with randomly oriented spins, therefore causing complete depolarization -194-of the muon spins, P = 0 . (II-6) res II.c. The ensemble of muon p o l a r i z a t i o n i n a single-step Mu reaction mechanism. The ensemble of muon p o l a r i z a t i o n at time t i s formally given by, P(t) = Z £ P (t) , (II-7) q q q where q represents the " f a t e " of a f r a c t i o n f of the muon ensemble, and P (t) gives the p o l a r i z a t i o n at time t of that f r a c t i o n . In a single-step Mu reaction mechanism, there are two types of f a t e s : Mu > Mu (unreacted) (II-8a) Mu > D . (II-8b) In (II-8a), muons i n s t i l l uncombined Mu atoms at time t ( i . e . free Mu atoms) have Pj(t) = e " U P D ( t ) , (II-9) where Pp(t) represents the complex y + p o l a r i z a t i o n i n free Mu atoms, PpC*) w i l l be given i n the next section. On the other hand, i n (II-8b), those muons i n diamagnetic products following r e a c t i o n at times t'<t have -195-} -At' - i " D ( t - f ) P 2 ( t ) = / Xe A P (t')e dt' . (11-10) o P(t) i s the sum of Pj ( t ) and P 2 ( t ) . In the l i m i t of t-*>°, ( i . e . a l l Mu atoms "eventually r e a c t ) , P j ( t ) = 0. Therefore, s u b s t i t u t i n g equation (11-10) into (II-3), one has an expression of P r e s f o r a single-step Mu reaction mechanism. » ,. -(A+ico n)t' P = A / P (t')e dt' . ( I I - l l ) res J o o I l . d . The general expression of P r e s f o r a l l f i e l d s . In order to evaluate the P r e s expression i n (11-11), Pp(t') must be cal c u l a t e d . According to the notations of Fleming et a l . [28], Pp(t) i s given by equation (11-12). P D ( t ) = -i{c (e + e ) + s (e + e )] (11-12) where: c = — [1 + ] ' (II-12a) /2 /l+x 2" 1 x h s = — [1 - — ] (II-12b) /2 /1+x2 and x = B / B Q . (II-12c) The frequencies, w 2, u>34, to 2 3 § a>14 are as defined i n chapter 2 (Figure 2.5). Substituting equation (11-12) into (11-11) and int e g r a t i n g gives the a n a l y t i c a l -196-expression f o r P r e s at a l l f i e l d s i n accordance with (11-13), M c 2 , c 2 , s 2_ r r e s 2 \ x + i ( u ) D - a ) 1 2 ) " A+i ( u ^ + w . ^ ) " A+i ( c d D - u > 2 3 ) ' (A+i ( a > D - u ) 1 4 ) (11-13) II.e. D i f f e r e n t f i e l d l i m i t s of P r e s Now one has an a n a l y t i c a l s o l u t i o n to p r e s f o r a single-step Mu reaction mechanism. Before evaluating equation (11-13), one should consider the d i f f -erent f i e l d l i m i t s and t h e i r implications f o r reactions of various rate constants. I I . e . l . B < 10G. At low magnetic f i e l d s (B<10G), x = 0, c = s = 1//2, ~ u23 ~ WM' a n d  w34 ~ w14 ~ w o ' e c l u a t ^ o n (H"13) becomes P = 4 [ —. ] , (H-14) res 2 A - iAw where Aco=cuD-u>M. For p r a c t i c a l a p p lications, the absolute magnitude of equation (11-14) can be rewritten as IP I = (P 2 + p 2yh , (n-15) 1 r e s 1 v x y 1 ' where P = (Re)P = ^ [ T T T T T ] . (II-15a) x res 2 L XZ + Au z -197-and P = (Im)P = y E ^ ^ A i ] • (U-lSb) y res 2 A ^ + A w -II.e.2. 10G<B<150G. In the case up to 150G where x < 0.1, equation (11-13) becomes = \ \ L,_ + „ X 2 2 ] (11-16) res 2 L A - i u ) M A z + C J Q Z noting that C J m >> ojp. By de f i n i n g P and P as i n (11-15), one obtains x y l P r e s l = I C a 4 + *V ) % / ^ + "M > ' ( I I - 1 7 ) There are several i n t e r e s t i n g but q u a l i t a t i v e features of equations (11-16) and (11-17). For extremely f a s t reactions ( A > > u ) o > > u ) ^ ) , P r e s > 1 ; (II-16a) that i s , a l l Mu atoms react at t =0 and therefore the p o l a r i z a t i o n has no chance to change from i t s i n i t i a l value. However, f o r normal reagent concen-t r a t i o n s ( A < < O ) Q ) , . 1 + ico Mx p > \ (.• 2 2 ) ; d i - i 6 b ) res 2 1 + iii?. x M -198-where T(=1/X) i s defined as the mean chemical l i f e t i m e of a thermal Mu atom. But when the reaction i s s t i l l much f a s t e r than Mu precession (WMT<<1), then (II-17a) whereas i f the opposite i s true ( i . e . U)^T>>1), then |P I > 0 . (II-17b) 1 res 1 7 These q u a l i t a t i v e features can be e a s i l y checked f o r one's Mu-reaction mechanism without going into any sophisticated (and t i r i n g ) c a l c u l a t i o n s with equation (11-13) . I I . f . Inclusion of Hot-atom reactions. So f a r one has only been t a l k i n g about those muons which thermalize as free Mu atoms. However, one must consider those epithermal Mu* atoms that -12 react at "hot-times" (-10 sec, t=0) and have no opportunity f o r d e p o l a r i -zation, or muons which do not form Mu. These muons contribute a constant unrotated f r a c t i o n h (as defined i n section 5.B.2) to the o v e r a l l r e s i d u a l p o l a r i z a t i o n , while the contribution (equation (11-13)) from thermal reactions of Mu atoms gives a f r a c t i o n of (1-h). Therefore, the o v e r a l l P r e s or P D i s given by equation (11-18). p n = p o c (overall) = h + (1-h) • P [equation (11-13)] . (11-18) -199-I l . g . A p p l i c a t i o n of P r e s to muon y i e l d s i n concentrated OH solutions. In order to a r r i v e at equation (5.9) i n section 5.B.2., one simply - substitutes the P expression f o r less than 150G (equation (11-17)) into 1*6 S equation (11-18) because A << u) Q. This gives the expression f o r the single-step Mu rea c t i o n in concentrated OH solutions as P D = h + 0.5(l-h) {A/(A 2 + t^ 2)"^} (H-19) PUBLICATIONS B.W. Ng, Y.C. Jean, Y. I t o , T. Suzuki, J.H. Brewer, D.G. Fleming and D.C. Walker, 1981, "D i f f u s i o n and a c t i v a t i o n -c o n t r o l l e d reactions of muonium i n aqueous so l u t i o n s " , J . Phys. Chem., v o l . 85, p. 454. (j) B.W. Ng, J.M. Stadlbauer, Y.C. Jean and D.C. Walker, 1983, "Muonium atoms In l i q u i d and s o l i d neopentane", Can. J . Chem., V.61, p.671. (?> B.W. Ng, J.M. Stadlbauer and D.C. Walker, 1983, "Muon spin r o t a t i o n involving muonium at high pH", J . Phys. Chem., i n press. (<9 B.W. Ng, J.M. Stadlbauer, Y. Ito, Y. Miyake, and D.C. Walker, "Muonium r a d i c a l s i n styrene-benzene mixtures", Hyper. Inter., i n press. (£) Y.C. Jean, D.G. Fleming, B.W. Ng and D.C. Walker, 1979, Chem. Phys. L e t t . , V.66, p.187. ( P Y . C . Jean, B.W. Ng and D.C. Walker, 1980, Chem. Phys. L e t t . , vol.75, p.561. (J)Y. Ito, B.W. Ng, Y.C. Jean and D.C. Walker, 1980, Can. J. Chem., V.58. p.2395. C£> Y.C. Jean, B.W. Ng, J.H. Brewer, D.G. Fleming, and D.C. Walker, 1981, J . Phys. Chem., vol.85, p.451. (5)Y. Ito, B.W. Mg, Y.C. Jean and D.C. Walker, 1981, Hyper-fi n e Interactions, vol.8, p.355. Q°> Y.C. Jean, B.W. Ng, Y. Ito, T.Q. Nguyen and D.C. Walker, 1981, Hyperfine Interactions, vol.8, p.351. (jj^Y.C. Jean, B.W. Ng, J.M. Stadlbauer, and D.C. Walker, 1981, J . Chem. Phys., vol.75, p.2879. QJ} J.M. Stadlbauer, B.W. Ng, Y.C. Jean, Y. Ito and D.C. Walker, 1981, Can. J . Chem., vol.59, p.3261. (ii) J.M. Stadlbauer, B.W. Ng and D.C. Walker, 1983, J . Am. Chem. S o c , vol.105, p.752. (i£> J.M. Stadlbauer, B.W. Ng, Y.C. Jean and D.C. Walker, 1983, J . Phys. Chem., vol.87, p.841. Q i^Y.C. Jean, B.W. Ng and D.C. Walker, 1982, i n "Applica-tions of Nuclear and Radiochemistry", p.543. J.M. Stadlbauer, B.W. Ng, Y.C. Jean, Y. Ito and D.C. Walker, 1983, i n " I n i t i a t i o n of Polymerization", p.35. (Yp J.M. Stadlbauer, B.W. Ng, Y.C. Jean, Y. Ito and D.C. Walker, 1983, Hyperfine Interactions, i n press. Miyake, Y. Tabata, Y. Ito, B.W. Ng, J.M. Stadlbauer and D.C. Walker, 1983, Chem. Phys. L e t t . , i n press. (L2)Y. Miyake, B.W. Ng, J.M. Stadlbauer, Y. Ito, Y. Tabata and D.C. Walker, 1983, "Temperature dependence of Muonium i n , hydrocarbons", Hyperfine Interactions, i n press. (jj>>J.H. Stadlbauer, B.W. Ng, R. Gantl and D.C. Walker, 1983, "Muonium addition reactions to aromatic solutes: a p p l i c a -t i o n of the Hammett equation i n the production of muonium-r a d i c a l s " , J . Am. Chem. S o c , submitted. 

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